As an undergraduate student whenever I studied some subject I tried to come up with problems. Many of these problems were artificial or silly and, of course, I forgot most of them. But a few still make sense. Here are two problems: 

1) Let B be the unit ball (or the unit cube) in $latex R^d$. Does every function from B to B which is the differential of a real function on B have a fixed point?

2) Is there a common generalization for Sylows's theorem and Frobenius's theorem?

Updates:A few typos corrected; thanks Lior! A remark by David Speyer suggests that both results we would like to find a common generalization to, are due to Frobenius. (Who was motivated by Sylow's theorems.) Emmanuel Kowalski's was partially motivated by these problems to present an old sporadic problem of his. (It is a mystery for me why his post is not mentioned as a track-back in this post.) 

1) A fixed point theorem for differentials?

One of the delightful theorems you learn in the first year of undergraduate studies (and later teach as a TA, and later teach as a professor) is the intermediate value theorem. If a continuous function satisfies $latex f(a)<0$ and $latex f(b) >0$ then for some point c in the interval [a,b], $latex f(c)=0$. Later you learn Darboux's theorem asserting that if f is a differential function, and $latex f'(a)<0$ and $latex f'(b) >0$ then for some point c in the interval [a,b], $latex f'(c)=0$. Your first reaction is: "Of course, $latex f'(x)$ is continuous and we can apply the intermediate value theorem." But, no, you soon learn some subtle examples where the derivative is not continuous!

One equivalent formulation of the intemediate value theorem asserts that every continuous map from [0,1] to itself has a fixed point. Namely, for some $latex x \in [0,1]$, $latex f(x)=x$. This is a special case of Brouwer's fixed point Theorem (which you learn in the second year) which asserts that every map from the unit ball of $latex R^n$ to itself has a fixed point.

We can ask if every function from the unit ball to itself, which is a differential of a real function on the unit ball, has a fixed point.

There are many very interesting and difficult problems related to basic real analysis. Not many mathematicians are fully aware of the rich and beautiful modern results in real analysis. (Just like many people outside mathematics are not aware that there is more to be discovered in mathematics itself.)

For example, it has only recently been proved by Csörnyei, O'neil, and Preiss and by Elekes, Keleti, and Prokaj, that the composition of derivatives of differential functions has the fixed point property. This is not easy at all. Also, the question regarding connectivity of the graph of differentials of functions was studied extensively. See this paper by Csörnyei and  Holický

In Budapest, I mentioned this problem to Miklos Laczkovich. (His UCL home page mentions a few open problems in real analysis.) He asked Marton Elekes (an author of one of the papers I mentioned above, and the son of György Elekes whom we mentioned in connection to product sum theorems). Elekes found a simple proof that the answer is yes - there is always a fixed point . Suppose you want to prove it for a function f(x,y) whose derivative maps the unit square into itself. What you need to do is to inspect the behavior of $latex f - x^2/2 - y^2/2$ in the boundary of the square.

So this problem was not so good, but the following problem proposed by Laczkovich might be. 

Problem: Let X be a set homeomorphic to the unit ball in $latex R^d$. Does every function from X to X which is the differential of a real function on X have a fixed point?

2) Joining Frobenius' and Sylow' theorems

Sylow's theorems in group theory, which we studied in the second year of undergraduate studies, always seemed to me as one of the few theorems I did not have a conceptual understanding of. This makes Sylow's theorems rather mysterious and charming. (A similar impression with the opposite reaction is expressed by Tim Gowers in this interesting post.) 

Sylow's theorem (one of them) asserts: In a group whose order is divisible by $latex p^i$ there are 1(mod p) subgroups of order $latex p^i$.

Frobenius' theorem asserts: In a group whose order is divisible n, the number of solutions to the equation $latex x^n=1$ is zero modulo n.

 (Frobenius was probably inspired by Sylow.)

Sylow intersection with Frobenius: The case i=1 of Sylow's theorem is the same as the case n=p of Frobenius' theorem.

Is there a nice (Sylow JOIN with Frobenius) theorem? The case i=2 of Sylow's theorem is the place to start.

Consider a family $latex \cal F$ of subsets of size d of the set N={1,2,...,n}.

Associate to $latex \cal F$ a graph $latex G({\cal F})$ as follows: The vertices of  $latex G({\cal F})$ are simply the sets in $latex \cal F$. Two vertices $latex S$ and $latex T$ are adjacent if $latex |S \cap T|=d-1$.

For a subset $latex A \subset N$ let $latex {\cal F}[A]$ denote the subfamily of all subsets of $latex \cal F$ which contain $latex A$. 

MAIN ASSUMPTION: Suppose that for every $latex A$ for which $latex {\cal F}[A]$ is not empty $latex G({\cal F}[A])$ is connected.

MAIN QUESTION:   How large can the diameter of $latex G({\cal F})$ be in terms of $latex d$ and $latex n$.

Our first lecture dealt with extremal problems for families of sets. In this lecture we will consider extremal problems for sets of real numbers, and for geometric configurations in planar Euclidean geometry. 

Problem I: Given a set A of n real numbers, how small can the set A+A={a+a': a,a' $latex \in$ A} be?

If A={1,2,...,n} |A+A|=2n-1. Suppose the elements of A are $latex a_1,a_2, \dots, a_n$ and $latex a_1< a_2<a_3\dots a_n$. Note that $latex a_1+a_1<a_1+a_2<a_1+a_2< \dots$  $latex a_1+a_n<a_2+a_n<\dots a_n+a_n$. So we identified 2n-1 distinct elements in A+A. This is the Cauchy-Davenport theorem.  

Problem II: Given a set A of n positive real numbers, how small can be the set A $latex \cdot$ A={a$latex \cdot$ a': a,a' $latex \in$ A}?

You may protest (again, like in the first lecture,) that I regard problem II a different problem. You can move from problem II to problem I by taking the logarithm of all elements in A, or you can simply use the same proof with the sum replaced by a product. The proof relies only on very basic monotonicity properties of these operations.

OK, lets have another problem II.

Problem II: Given a set A of n real numbers, how small can the quantity max (|A+A|, |A $latex \cdot $ A|) be? 

This problem was asked by Erdös, and in hindsight it is a very good problem. Erdös conjectured that the maximum behaves like $latex n^2$, and this is open.  We will see below a wonderful proof by Elekes that the maximum is (up to a multiplicative constant) at least $latex n^{5/4}$. The exponent 5/4 was improved twice(!!) by Jozsef Solymosi and there is a very nice post about his most recent ingenious proof for a lower bound max (|A+A|, |A $latex \cdot $ A|) $latex \ge$ $latex (1/2) n^{4/3} (\log n)^{-1/3}$ in Izabella Laba's blog. 

Extremal problems in plane geometry

Problem III:  Given n points in the plane what is the maximum number of pairs among them at distance '1'

Problem IV:  Given s points and t lines in the plane what is the maximum number of incidences between them.

An incidence is a pair $latex (p,\ell )$ where $latex p$  is a point $latex \ell$  is a line and $latex p \in \ell$.

Problem V: Given a graph G with v vertices and e edges what is the minimum number of crossings in a planar drawing of G.

The second of my extremal combinatorics lectures was devoted to the surprising connections that were found between the above problems. There is also a very nice post about this material on Terry Tao's blog. To show you something new I will describe a few problems in higher dimensions at the end.

The solution to problem IV is given by:

The Trotter Szemeredi Theorem: The number I of incidence between t points and s lines in the plane is at most $latex K \max (t,s,t^{2/3}s^{2/3})$.

In particular the number of incidences of n points and n lines in the plane is $latex O(n^{4/3})$. This is an Euclidean phenomenon which does not follow from the very basic axioms of geometry about incidences of points and lines: Recall that for a finite projective plane with n lines and n points the number of incidences behaves like $latex n^{3/2}$.

Elekes: Bounds for product sums via Trotter-Szemeredi

This is an example of an ingenious, simple, and surprising connection between two areas.

Here is how it goes: Let $latex A$  be a set of n positive real numbers. Consider the planar set $latex Z= (A+A) \times (A \cdot A)$. The number $latex t$ of points in this set is the quantity we would like to estimate. Now, consider the set of  lines of the form $latex y=(x-a_1) \cdot a_2$, where $latex a_1,a_2 \in A$. Our set of lines contains $latex s$ lines, $latex s=n^2$. Note that for every line of the form  $latex y=(x-a_1) \cdot a_2$ and every $latex a_3 \in A$ if we let $latex x=a_1 +a_3$ we get $latex y=a_3 \cdot a_2$. Thus the point $latex (a_1+a_3,a_2\cdot a_3) \in Z$ is on the line. We have identified $latex n$ points of Z on every line in our family of lines. In sum, the number of incidence is at least $latex n^3$.

Aha! But now we can apply Trotter Szemeredi's theorem: $latex n^3 \le K \max (t, n^2, t^{2/3} n^{4/3})$. This gives $latex t^{2/3} \ge K' n^{3-4/3}$ so $latex t \ge n^{5/2}$.        

In order to prove the Trotter-Szemeredi theorem we will need the following:

The Crossing Theorem: A graph drawn in the plane with v vertices and e edges, e>5v has at least $latex 1/70 e^3/v^2$ crossings.

Szekély: Bounds on Incidences via the crossing lemma

This is an example of a surprising ingenious and simple connection in the same area.

Here is how it goes: We can assume that every line is incident to a point. Consider the following graph drawn in the plane. The vertices are the points in our family. The edges correspond to line segments between two consecutive vertices on the same line. Now, this graph has $latex t$ vertices and $latex I-s$ edges. (A line incident to k points contributes k-1 edges.) The number of crossings is at most $latex s \choose 2$ because every crossing represents an intersection point between a pair of lines. Therefore either $latex I-s \le 5t$ or $latex t^2 \le 1/70 (I-s)^3/t^2$.  

The bootstrap proof of the crossing lemma

This is an example of a simple new proof that sheds a light on a theorem, yet is it really different from the old proof? (There was an interesting discussion over Gowers' blog on when two proofs are essentially the same. In this case, I regard the proofs as different while Janos Pach who understand this subject more deeply regards them as the same.)

We start with Euler's theorem. It implies that a planar graph with v vertices has at most 3v-6 edges. Therefore for a graph G with e edges and v vertices drawn in the plane consider a maximal planar subgraph H and note that every edge not in H crosses an edge in H. So  $latex c \ge e-3v+6$ .

And now we improve this inequality using itself as follows: Start with a graph G with v vertices and e edges, e>5v. Choose at random every vertex with probability p. The resulting random graph H will have v' verices, e' edges, and c' cossings where the expectation of v' is pv, the expectation of e' is $latex p^2e$ the expectation of c' is $latex p^4 c$. It follows that $latex p^4 c \ge p^2 e - 3 p n$ so we can optimize the value of p and this leads to the crossing lemma.   

Problem III about the number of pairs of points at distance one among n points is the plane is a very famous problem (of Erdös, of course) in discrete geometry. Consider a configuration of n points in the plave and the configuration of unit circles around them. The number of pairs of distance one is half the number of points-circles incidences. Trotter and Szemeredi proved an upper bound $latex K n^{4/3}$ and Szekely's argument gives a simple proof of this result as well. It is conjectured that the exponent 4/3 can be pushed down to $latex 1+\epsilon$, for every $latex \epsilon$. The Trotter-Szemeredi theorem itself is sharp - there are examples with matching lower bounds. (Up to the multiplicative constant K.)

High dimensions

Conjecture: Let K be a 2-dimensional simplicial complex with e edges and t triangles. Then for every embedding of K into $latex R^4$ the number of crossing is at least $latex C t^4/e^3$.

As pointed out by Dey and Pach, this conjecture would follow (using the probabilistic argument above) from the following:

Conjecture (Sarkaria): Let K be a 2-dimensional simplicial complex with  $latex f_2(K) \ge 4 f_1(K)$ (or even with $latex f_2(K) \ge 4 f_1(K)-10 f_0(K) +20$) then K cannot be embedded to $latex R^4$.

(There are similar conjectures for higher dimensions.)

Sarkaria's conjecture is closely related to the "g-conjecture" that I plan to devote a special post to.

Problem: What is the maximum number of incidences between s lines and t planes in $latex R^4$?

Unlike the planar case I do not know of a relation between the incidence problem and the crossing problem in high dimensions.

Economic puzzles told by Arrow

Let me tell you about three economics puzzles mentioned by Arrow in an earlier summer school. I doublechecked some details with Arrow himself; still, if my description contains errors I will be happy to be corrected. (Arrow spent a considerable amount of time talking with the workshop students. Another remarkable thing about him: he takes lecture notes! Is it a good idea to take detailed lecture notes at lectures? Let's return to this question sometime.)

Puzzle 1: Why is there unemployment?

Why is this even a puzzle? Because the economics teaching that "the market will clear" means that all people who can work will. A person who can work and is not working represents inefficiency, which is not supposed to exist in a competitive economy. Part of the issue is referred to as "friction" and accounts for economics processes being slow rather than instantaneous. But it appears to be true that there is more to unemployment than that. What can explain the 30% unemployment that was witnessed in the US in the 1930s?

Is this puzzle a scientific problem? You bet it is! And it is a fairly clear-cut scientific problem. I suppose there are several answers to this puzzle in the literature but we are far from a definite understanding of the issue.

Puzzle 2: What is the reason for high volatility of prices in markets, say in stock markets?

The price of a stock, according to economics theory, represents the long-term value of the company. What accounts for the fact that the overall value of the entire stock market may fluctuate by more than 1% on a typical day? What accounts for fluctuations (more often drops) of 3-5% in one day? (Such fluctuations are not rare.) A famous question is to explain the one-day drop of 20% in October 1987.

Arrow mentioned in this context the Milgrom-Stokey "no trade theorem" which asserts that under certain assumptions markets in equilibrium will exhibit no trade (even if traders have private information).

Private companies conduct a lot of research on stock market behavior, probably much much more so than universities. I asked Arrow whether we should expect some progress toward understanding the fundamental issues regarding stock-market behavior to be achieved there. Arrow was quite skeptical about it. 

In my opinion, stock market behavior is an example where scholarly research is important even in areas where much research is taking place outside academia. (It is also an important and delicate matter to ensure that the external research and activity not vitiate academic goals and integrity.) 

A relevant blog post concerning financial mathematics is Tao's recent description of the Black-Scholes formula. Explaining the systematic difference between the Black-Scholes formula and the actual behavior of options prices is another interesting question.

Puzzle 3: What accounts for the huge futures trading in foreign currencies?

Another puzzle that Arrow mentioned is this: futures trading in foreign currencies can be explained by agents involved in international trade wanting to reduce their risk. This suggests that the volume of currency futures trading  will be below the volume of international trade. Yet currency futures trading is 300 times larger. What can explain this phenomenon?

Following are a few lectures that I would like to tell you about, in some detail, in a later post. Maskin’s lectures on social choice and “the robustness of majority rule” were perhaps the lectures closest to my own research interests (and related to Arrow’s theorem about voting). Roger Myerson in his lecture asked: "Is capitalism better than socialism?" He was referring to the Soviet Union-type of socialism and the way firms operate under such a system. John Geanakoplos talked about models of general equilibrium theory with collateral and gave a hilarious account of Shakespeare as economist. His model provides some insight into the current subprime crisis in the US.  And Herb Scarf returned to cooperative game theory and proved his theorem on the nonemptyness of the core for balanced games. A link to all lectures is here. 

Apropos of the comparison between capitalism and socialism, Myerson's work does not deal with a comparison between the US system and the slightly more socialist West European version. (The difference does not lie in the way firms operate but in governmental redistribution of resources.) Personally, I like the West European economic free-market system and even the most "socialist," Scandinavian version of  it, and this once almost got me in trouble. When I visited Stockholm, in the late 80s, I was sitting next to a local Swedish person in a Chinese restaurant on Nybrogatan Street, and I was telling him at some length how highly I thought of the Swedish system. The guy listened carefully to what I said and at the end he was so angered by it that I thought he would kill me. (In Sweden, at that time, the punishment for murder was 10 years in prison, of which only five had to be served; yet murder rates were low.) It appears that one should be careful about giving compliments almost as much as about criticism.

11. How to make it work - the matrix tree theorem

Our high dimensional extension to Cayley's theorem reads:

$latex \sum |H_{d-1}(K,{\bf Z})|^2 = n^{{n-2} \choose {d}},$

where the sum is over all d-dimensional simplicial complexes K on n labelled vertices, with a complete (d-1)-dimensional skeleton, and which are Q-acyclic, namely all their (reduced) homology groups with rational coefficients vanish.  

Looking at the various proofs of Cayley's formula (there are many many many beautiful proofs), the proof that I know to apply is the one based on the matrix tree theorem.

Consider the signed incidence matrix A' between all (d+1)-subsets and all d-subsets of {1,2,...,n} that represents the boundary operator of simplicial homology. The rank of this matrix is $latex {n-1} \choose {d}$, and just like in the ordinary matrix tree theorem you delete rows to be left with linearly independent rows. Here you delete all rows corresponding to sets containing 'n' and you are left with a matrix A. Now we compute the determinant of $latex det (A \cdot A^{tr}) $ directly, and compare the result to a computation based on the Cauchy-Binet Formula.

The eigenvalues of $latex A \cdot A^{tr}$ are the eigenvalues of the d-th Laplacian of the complete d-dimensional simplicial complex with n vertices. It is easy to inspect what they are and the determinant of $latex A \cdot A^{tr}$ is indeed $latex n^{{n-2} \choose {d}}$.

The many square determinants correspond to d-dimensional simplicial complexes K on our labelled set of vertices, which satisfy $latex f_{d-1}(K) = {{n} \choose {d}}$ and $latex f_d(K) = {{n-1} \choose {d}}$. Now if K has non-vanishing d-th homology, the determinant is zero. If K is a Q-acyclic simplicial complex (i.e., its (reduced) homology groups with rational coefficients are trivial) then it has a non zero determinant. So far, it is like trees, but next comes a surprise. The contribution of K is the square of the number of elements in $latex H_{d-1}(K)$, the (d-1)th homology group of K. This torsion group is finite, but for d >1 it need not be trivial.

Emily Peters presents the matrix-tree theorem. From "The Sarong Theorem Archive" - an electronic archive of images of people proving theorems while wearing sarongs.

12. An even simpler use of Cauchy-Binet worth knowing

Consider the $latex n \times 2^n$ matrix A, whose columns are all +1 -1 vectors of length n. Computing $latex det A A^{tr}$, via Cauchy-Binet Formula (or by other easy methods) asserts that the expected value $latex (det (B))^2$ for all $latex n \times n$  +1/-1 matrices behaves roughly like $latex (n!)$. This was observed by Turan and Szekeres who also found a formula for the sum of the fourth powers of all 0-1 n by n matrices. See a leter paper by Turan. (I am not aware of a similar formula for the fourth power of the size of the homology groups for hypertrees.) Much is known about the determinant and related properties of random 0-1 matrices and the analogy between torsion in the homology groups of random complexes and determinants of random matrices looks like a good analogy.

13. Torsion

One consequence of the formula compared to the total number of available simplicial complexes is that the torsion group is typically huge. (For d>1, the expected value of $latex |H_{d-1}(K,{\bf Z})|^2$ for Q-acyclic complexes counted by the formula, is asymptotically larger than $latex ((d+1)/e)^{{n-2} \choose {d}}$; I am not aware of explicit examples with such a huge torsion group.)

How should we think about torsion in homology? It seems that thinking about the size of the torsion as a behaving like the determinant of a random matrix, may give a good intuition for many cases.

14. Extending other proofs for Cayley's theorem?     

Cayley's counting trees theorem has many wonderful proofs.  Can any other proof extend to the case of Q-acyclic simplicial complexes? For example, one proof relies on the exponential theorem that relates the exponential generating functions for connected and general graphs with a certain property P. (Followed by the Lagrange inversion formula.) Is there an analog of the exponential formula when connectivity is replaced by higher homology? Is there any analog of Prüfer sequences? I am not aware of any other proof that works.

15. Weights to the rescue of other conjectures? 

Can we use subtle weights to save other promising but false enumerative conjectures? The farthest reaching fantasy in this direction would be to try to save MacMahon's conjecture regarding space partitions of the number n. This conjecture is about enumerating spacial arrays of numbers that sum up to n. The conjecture is true for small values of n but fails for larger values. Can subtle weights come to the rescue?  (MacMahon's conjecture extends the formulas for ordinary partitions and for plane partitions.) 

16. Incidence matrices

Perles' observation in the beginning of this story was about the rank of the incidence matrices modulo 2 of k-subsets versus (k+1) subsets of an n element set. This was the starting point for a work by Linial and Rothschild. They asked:  What is the rank of the incidence matrix of $latex N \choose r$ versus $latex N \choose k$ modulo p? and gave a complete answer for p=2. Richard Wilson gave a complete answer for general values of p and came quite close  to presenting the "Smith form" of these matrices. Frumkin and Yakir gave representation-theoretic interpretation of Wilson's result and proved "q-analogs". Namely they replaced k-subsets of an n-element set by k-dimensional linear (and in a later work affine) subspaces of an n dimensional linear space over a field with q elements. They gave a complete formula when p and q are prime.   

17. Duality and Self-Dual trees

 Here is a very nice notion of duality that occurs in many places. Start with a simplicial complex K on a set X of vertices. Take the family F of all the complements to all sets in K. (This is not a simplicial complex, it is closed under supersets and not under subsets.) Now, take the family K* of all  subsets of X not in F. Formally, $latex K^*=2^X \backslash$ {$latex X \backslash S: S \in K$}.

K* is a simplicial complex again. It is the Alexander dual of algebraic topology, and the "blocker" of polyhedral combinatorics.

If n=2d+2 the duals of our hypertrees are also d-dimensional.  Molly Maxwell counted self-dual hypertrees with the same weights we used, and for odd dimensions the count gives precisely the square root of $latex n^{{n-2} \choose {d}}$. She deduced it from a more general theorem on matroids duality. For even values of d this is not the case but something may still work.

For d=1 the number of self-dual trees (simply stars) on 4 vertices is four the square root of 16 the total number of labelled trees. There is a theorem of Tutte extending this to self dual trees inside self dual planar maps. For d=3 and 8 vertices, the weighted number of all hypertrees is $latex 8^{20}$ and by Maxwell's theorem the weighted number of self dual ones is $latex 8^{10}$. For d=2, n=6 - the weighted number of hypertrees is $latex 6^6$ and if we exclude the triangulations of the real projective plane we get $latex 6^3$. For d=4 the total weighted sum of hypertrees with 10 vertices is $latex 10^{70}$, and somehow, a clever weighted sum of the self dual ones should give you $latex 10^{35}$.

18. The Perles-Katchalski conjecture and associated eumeration problem

The assertion of the Perles-Katchalski conjecture holds for general classes of simplicial complexes described by homological properties, and we can ask again if the extremal examples enumerate nicely.

Let $latex \cal K$ be the class of (d+r)-dimensional simplicial complexes with the properties that

(L) For every induced subcomplex K', $latex H_i(K',Q)=0, i \ge d$

Now, the homological extension of the Perles-Katchalski Theorem asserts that a (d+r)-dimensional simplicial complex K with n vertices satisfying condition (L) has at most  $latex {{n} \choose {d} } - {{n-r-1} \choose {d}}$ d-dimensional faces.(For r=0 we need not worry about induced subcomplexes since, in this case, non trivial d-th homology for a subcomplex immediately extends to the whole complex. 

We can try to "enumerate" simplicial complexes with n labelled vertices satisfying property (L) with precisely $latex {{n} \choose {d} } - {{n-r-1} \choose {d}}$ d-dimensional faces. (All these complexes will have the same number of i-faces for every i.)

We can expect that an appropriate enumeration of these objects (probably those containing a specific r-face), will give us a formula of the form $latex m^{{n-2-r} \choose {d}}$, where m is the number of r-dimensional faces of such a simplicial complex K. For the case d=1, no weights are needed and this speculation reduces to a formula of Beineke and Pippert for counting "k-trees". (We may even expect finer enumeration formulas according to degree-sequences of r-faces; This is known for "k-trees". See the following paper by C. Rényi and A. Rényi.)

19. Adin's colorful extension.

Ron Adin extended the weighted enumeration of hypertrees to "colored complexes", thus confirming (with extra weights added) another conjecture of Bolker.

20. Gelfand's question.

Ron Adin gave a lecture about his work in a Stockholm '89 meeting in combinatorics which was one of the earliest meetings with many participans from Russia, among them Gelfand, Vershik, Zelevinskii, Serganova, and others. Gelfand was excited about combinatorics (or what he regarded as combinatorics) at the time and was quite interested in Adin's result. One question he asked me was: why is it that in combinatorics there is so much emphasis on graphs compared to higher dimensional objects.

I personally like the combinatorics of high dimensional objects but I could think of three answers. (Gelfand was quite satisfied with them).

a) For many purposes moving from sets to graphs represents a major conceptual jump, more than moving up from graphs to higher dimensional objects.

b) Higher dimensional objects can often be represented by graphs.

c) Many of the miracles of graph theory fail at higher dimensions.

Another memory from the 89 conference is this: Israel Gelfand has a somewhat wide-spanned competative nature. Gelfand looked at Ron Adin and asked me: "He is orthodox isn't he?", "yes" I replied. Gelfand thought a little and then said: "But not as orthodox as my Dima."

It is quite interesting to compare this problem with the issue of the size of an unbalanced product of two $latex SL(2,\mathbb{R})$ (already discussed in this blog). Basically, these problems (as well as the partial results of Kaloshin-Rodnianski and Fayad-Krikorian) are very similar in spirit, as the reader can infer from the statements and the corresponding arguments.

That's all folks! See you!

When do we say that one event causes another? Causality is a topic of great interest in statistics, physics, philosophy, law, economics, and many other places. Now, if causality is not complicated enough, we can ask what is the influence one event has on another one.  Michael Ben-Or and Nati Linial wrote a paper in 1985 where they studied the notion of influence in the context of collective coin flipping. The title of the post refers also to Nati's influence on my work since he got me and Jeff Kahn interested in a conjecture from this paper.  

Influence 

The word "influence" (dating back, according to Merriam-Webster dictionary, to the 14th century) is close to the word "fluid".  The original definition of influence is: "an ethereal fluid held to flow from the stars and to affect the actions of humans." The modern meaning (according to Wictionary) is: "The power to affect, control or manipulate something or someone."

Ben-Or and Linial's definition of influence

Collective coin flipping refers to a situation where n processors or agents wish to agree on a common random bit. Ben-Or and Linial considered very general protocols to reach a single random bit, and also studied the simple case where the collective random bit is described by a Boolean function $latex f(x_1,x_2,\dots,x_n)$ of n bits, one contributed by every agent. If all agents act appropriately the collective bit will be '1' with probability 1/2. The purpose of collective coin flipping is to create a random bit R which is immune as much as possible against attempts of one or more agents to bias it towards '1' or '0'.

Given such a protocol, the influence of a set S of agents towards '0' is the probability that R=0 if the agents in S try to tilt the outcomes of the coin flipping towards '0' as much as possible. The influence towards '1' is defined in the same way. And the influence of S is the sum of these two quantities. To make the definition clearer we should explain what the agent in S can do. Here we assume that in case of simultaneous action by all agents, the "bad guys" can wait to the contributions of all other agents before making their move. The bad guys can only change their inputs to the procedure.

Notations: When the protocol is denoted by $latex f$, for a set $latex S$ of processors, denote by $latex I^+_S(f)$ their influence toward '1', by $latex I^-_S(f)$ their influence toward '0', and let $latex I_S(f)= I^+_S(f)+I^-_S(f)$. The influence of a single processor $latex k$ is denoted by $latex I_k(f)$ and the sum $latex I(f) = I_1(f)+ I_2(f)+ \dots +I_n(f)$ is called the total influence of $latex f$. (In the Boolean case we refer to a "processor" or "agent" simply as a variable, and talk about influence of a variable, and influence of a set of variables.)

Notions of powers 

Boolean functions can be regarded as "voting rules". A Boolean function describes a way to move from the votes of n voters between two candidates to the collective decision of the society. Thinking of Boolean functions as voting rules provides nice names for special kinds of Boolean functions. "Dictatorship" refers to functions of the form $latex f(x_1,x_2,\dots, x_n)=x_k$. For the "majority function" (when $latex n$ is odd) the value of $latex f$ is '1' if and only if for more than half the variables $latex x_k =1$. For monotone Boolean functions, the influence of the kth variable coincides with the "Banzhaf power index" defined in game theory. Another related important notion of power is the Shapley-Shubik power index.    

The KKL Theorem

Picture: Muli Safra

After many months of working on the conjecture, Jeff, Nati, and I managed to prove it. Let $latex f(x_1,x_2,\dots,x_n)$ be a Boolean function.

Theorem 1 (KKL): If the $latex \mu (f)=t$ then there exists a variable k so that $latex I_k(f) \ge K t (1-t) \log n/n$.

A repeated application of this theorem shows that:

Theorem 2 (KKL): If the $latex \mu (f)=1/2$ then there exists a set S of $latex K(\epsilon) n/\log n$ variables so that $latex I_S(f) \ge 1-\epsilon $.

Examples, examples

Majority

For the majority function with n variables the influence of every variable is proportional to $latex \sqrt n$.  

The tribes example

This is the basic example of Ben-Or and Linial for Boolean functions with low influence. The society is divided into a large number of tribes, each having $latex \log n - \log \log n +\log \log e$ members. The value of f is one if and only if there exists a tribe whose members all vote '1'. For this example the influence of every variable is $latex \theta (\log n /n)$.  

 The next two examples represent more complicated protocols for collective coin flipping. (Not just Boolean functions.) 

Mike Saks' "passing the baton" example

We start with some voter who holds the baton. This voter passes the baton to another random voter. Every voter who gets the baton passes it to a random voter who did not yet hold it. The last voter to hold the baton chooses the random collective bit. In this example, $latex o(n/\log n)$ bad agents cannot tilt the outcome significantly. (The best strategy for the "bad guys" is to pass the baton to a "good guy".)

Uri Feige's "two rooms" example 

Every agent enters at random one out of two rooms. The room with fewer agents is selected and every agent in this room enters at random one out of two rooms. This process is continued (more or less) and at the end, as before, the last remaining agent contributes the collective random bit.

This process is immune against a constant number of "bad guys". (The first such example was found by Noga Alon and Moni Naor.) The number of rounds in this protocol (appropriately optimized) goes to infinity extremely slowly. It is not known whether there is a protocol with similar properties with a bounded number of rounds. 

Influence and threshold behavior

Consider a monotone Boolean function $latex f(x_1,x_2,\dots,x_n)$. Let $latex \mu_p$ be the product probability space where for every bit $latex \mu_p(x_i=1)=p.$ The definition of influence extends without change to the setting of biased product distribution. The influence of the $latex k$th variable on the Boolean function f with respect to $latex \mu_p$ is denoted by $latex I_k^p(f)$. The probability $latex \mu_p (f)$ that $latex f(x_1,x_2,\dots,x_n)=1$ is a monotone function in $latex p$ and Russo's lemma asserts that the derivative of $latex \mu_p (f)$ with respect to $latex p$ is precisely the total influence $latex I^p(f)$. Therefore, large influence is related to "sharp threshold behavior". Namely, to a very short interval between the value of $latex p$ where $latex \mu_p(f)$ is very close to 0, and the value of $latex p$ where $latex \mu_p(f)$ is very close to 1. Simple consequences of KKL's theorem to the study of threshold behavior were noted by Ehud Friedgut and me, and Friedgut found an important theorem giving conditions for sharp threshold behavior when p itself is a function of n. Muli Safra and I wrote a survey paper on influences and threshold behavior. 

Aggregation of information and Condorcet's Jury theorem

Four sentences about the connection with Game Theory: The sharp threshold phenomenon is called in economics "asymptotically complete aggregation of information". This property goes back to an old theorem from the theory of voting called "Condorcet's Jury Theorem". The Shapley-Shubik power index, mentioned above, can be defined as the integral $latex \int_0^1 I_k^p(f)dp$.  (This is not the original axiomatic definition but a later Theorem by Owen.) It turns out that for a sequence of monotone Boolean functions, "sharp threshold phenomenon" is equivalent to "diminishing individual Shapley-Shubik power indices". (But the quantitative aspects of this result are not satisfactory.)     

Influence without independence

A major conceptual challenge is to understand the concept of influence (and related notions of "sharp threshold phenomenon") for distributions which are not product distributions, namely when the probabilities for individual bits to be '1' are not statistically independent. Does an observer in a committee meeting have an influence? Can there be a negative influence? Moving away from statistical independence is often very difficult and yet very important for most applications. This issue is addressed in the paper of Graham and Grimmett, and that of Haggstrom, Mossel, and myself.

Two old conjectures about influence

There are quite a few problems regarding influences which remained unsolved. I will mention only two related conjectures both dealing with the Boolean case.

Conjecture 1: (Benny Chor): Suppose that $latex \mu(f)=1/2$ there is $latex c>0$, such that there is a set $latex S, |S|= n/10$  with $latex I_S(f) =1-\exp (cn)$.

Conjecture 2: Suppose that $latex \mu(f) = 0.999^n$. Then there is a set $latex S, |S|=0.499n$, so that $latex I_S(f)=1-o(1)$.

KKL theorem gives a weak form of Conjecture 1 where $latex 1-\exp (cn)$ is replaced by $latex 1-n^{\beta}$ and a weak form of Conjecture 2 where $latex 0.999^n$ is replaced by $latex 2^n/ n^{\alpha}$. The main difficulty here (and in various other problems in extremal combinatorics) is that arguing about influences of single variables is the only known method toward influences of large sets. Both these conjectures (as KKL's theorem itself) have natural formulations in terms of traces of families of sets related to the Sauer-Shelah Theorem.

More on Influence

The theory of poetic influence was developed by Yale's literary critic Harold Bloom. His book "The Anxiety of Influence" deals with the process of influence as well as with the psychology of influence in literature. Philosopher Avishai Margalit studied influence in Philosophy in his paper on Wittgenstein. The paper will appear in a Festschrift for Peter Hacker by Blackwell Oxford, 2009. Section 3 opens with a distinction between influence and power: "Naked power is for anyone to see. Influence is not. It works its wonders in ways not readily observable. Influence is inferred from its effects. This is a major reason for the elusive nature of influence."

Michael Ben-Or 

Trivia question:

What do Michael Ben-Or, Uzi Segal (whose calibration theorem was mentioned in the post on the controversy around expected utility theory), Mike Werman, Ehud Lehrer, Yehuda Agnon, myself, and quite a few others, all have in common.

Hint:

Influence and coin flipping little updates (June, 8)

Mike Saks made valuable comments regarding this post. The first protocol for collective coin-flipping immune to a positive proportion of processors was by protocol with $latex log^*n$ rounds was by Alexander Russell and David Zuckerman. If each processor contributes only one bit per round there is a matching lower bound by Russell, Saks, and Zuckerman. The open problem Section from RSZ's paper is very interesting and, as far as we know, no further progress on the problems presented there was made.

If we allow every processor to contribute many bits in every round then the situation is not clear even for one round. This is related to conjectures discussed by Ehud Friedgut.  

Related problems are of great interest for quantum computation. Carlos Mochon have recently solved one of the most important longstanding open problems in quantum information theory, and found a protocol for weak coin flipping with arbitrarily small bias, using quantum bits. His work strengthens earlier work by Kitaev.

Related post: The Entropy Influence Conjecture. (Terry Tao's blog) 

Local Events

The second semester at HU started on Sunday, May 11th and it will run until August. This is due to the 3-months Israeli Professors' strike at the beginning of the academic year. Issues regarding the strike and Israeli academics are quite interesting and we may come back to them. Let me make just one little remark: There is an initiative to transform Israeli universities to a more "market-based" structure. US universities and the new evaluation system in the UK are mentioned as examples, and the Australian academic reforms are often regarded as an act to follow. I was always quite negative about this initiative and skeptical even about the Australian example, and the following post by Terry Tao is telling regarding the Australian reforms. (See also the new blog mathematics in Australia.)

Thia semester I am teaching the basic course in combinatorics and a seminar in probabilistic combinatorics. As usual, things are hectic here. Our yearly spring school "Midrasha Matematicae" is devoted this year to Cluster algebras, quantization and higher Teichmuller theory. Cluster algebras were invented by Sergey Fomin and Andrei Zelevinski (both are visiting) and the starting points were questions in combinatorial representation theory and total positivity. In combinatorics they led, among many other things, to nice results regarding "Somos sequences" and to new extensions of the associahedron (=Stasheff polytopes). There is a lot of activity regarding cluster algebra related to geometry, algebra, categorization, tropicalization, and other high flight mathematics.

Beside that, we have our usual Monday's combinatorics seminar, (yesterday - Doron Zeilberger), and Wednesday's CS theory seminar, and Thursday's quantum computing seminar (with some extra lectures this week by Daniel Gottesman), and the colloquium, and the "basic notions" seminar, (and the Amitsur algebra seminar, and PDE seminar,) and Tuesday's Dynamics and probability seminar, and Sunday's game theory seminar, and the "Rationality on Friday" seminar (and a few more).  The first week also started with Avi Wigderson giving a great popular lecture on the "digital envelope" and its uses. He gave a lovely colorful (literally) explanation for zero-knowledge proofs.

Aner's colloquium on the Ore Conjecture was a wonderful mixture of various issues. The proof (what was "left" to be proved after earlier works was proving the conjecture for finite simple groups of Lie type over fields of less than eight elements), relied on deep representation theory (detailed understanding of Deligne-Lusztig characterization of representations of these groups), complicated inductive procedures, and three years of CPU time based on complicated computational group theory to give the necessary basis for the other methods. (Of course, it relies on the classification theorem for finite simple group.) ... I am not sure if I will continue with real-time blogging like this. (Little update (May 22): I forgot to mention talks by two fresh Wolf prize laureates; a beautiful and clear talk by Griffiths on the Hodge conjecture yesterday; Deligne will speak next week.)

Turan's  problem

In the post regarding extremal combinatorics I left it to the readers to guess a conjecture on collections of triangles without a "tetrahedron"; Let's do it together. Then I will give a few more details about a lecture by Szegedy from the Marburg conference , on graphs and hypergraphs limits. These two topics are not entirely unrelated. 

Let's try to think together what can be an example of a 3-uniform hypergraph (collection of triples) on n vertices N={1,2,3,...,n} without a "tetrahedron" - {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d},  with as many triangles as possible:

Step 1: So to start we can divide our vertices to 3 as equal as possible parts A, B and C, and take all triangles {a,b,c}. This gives roughly 2/9 of all triangles. It does not contain even three triangles of a tetrahedron: we can do better. (To find the maximum number of triangles without having three triangles spanned on four vertices ia also a very nice problem.)

Step 2: OK, let's divide N to two equal as possible parts A and B, and take all triangles of the form a a' b and a b b'  --- We got many triangles but this construction is not good, we have a tetrahedron a a' b b'. We can keep this example to another problem.

Step 3: OK, divide N into equal parts A and B and take only triangles of the form {a,a',b}. This gives ... 3/8 of all triangles, an improvement. 

Step 4: Hmm, we can now optimize the sizes of A and B. If we take A to contain 2/3 of the vertices, we get roughly 4/9 of all triangle.

Step 5: We can apply the "symmetry-breaking" of step 3 to the original construction of step 1. We can divide our vertices to three as equal as possible parts A, B and C and take all triangles {a,b,c} {a, a, 'b}, {b,b'c}, and {c,c'a}.  This is Turan's example. Turan conjectured in 1940 that this is the best example "on the nose" and his conjecture is still open.

When asking students to come up with examples of 3-uniform hypergraphs without tetrahedra, they often start with Step 1 and sometimes move directly to the right (conjectural) answer, and sometimes go through examples like the one in step 3 or try to partition the ground set into four parts.

I do not know how Turan reached his conjecture. Turan's theorem and Turan's problem, as well as the some basic problems regarding crossing numbers of complete bipartite graphs and complete graphs, were discovered when Turan was in a labor camp during World War II. He could not use pen and paper, so doing combinatorics was easier than working on analytic number theory which was his main area of research. Vera Sos (who is visiting here) referred me to Turan's welcome note in the Journal of Graph Theory (J. Graph Theory 1(1977) 7-9), which includes some memories of his graph theory research in the labor camps.  It starts with: "It sounds a bit incredible but it is true," and it is quite incredible indeed!

An amazing fact is that there are many examples by Brown and by Kostochka which give precisely the same number of triangles as Turan's original example. 

I will be happy to further discuss Turan's problem. Here is one small observation.

Can there be an even simpler theorem about triangle-free graphs than Turan's theorem?

When we move to the complement graph, Turan's theorem tells us that the minimum number of edges in a graph without an independent set of three points is obtained by the union of two complete graphs of equal as possible sizes. Here is something even simpler:

A graph with no three independent vertices has at most two connected components.

This very simple fact does generalize to:

A three uniform hypergraph H on n vertices so that every four vertices span a triangle satisfies: $latex \dim H_1(H',k) \le n-2$ .

Here $latex H_1(H',k)$ is the first homology group over an arbitrary field of coefficients $latex k$, and H' is the simplicial complex on N with a complete graph as the 1-dimensional skeleton, whose 2-faces are the set of our triangles.   

I do not know a "conceptual" proof for this fact. (Such a proof may lead to extensions toward Turan's problem.)  Homology inclined readers are welcomed to try. 

Limits of graphs and hypergraphs

Let me mention one talk from Marburg. When do we say that a sequence of graphs $latex G_n$ (or of hypergraphs) has a limit? First we need the important notion of graph homomorphisms.

A map between the vertices V(G) of a graph G and the vertices V(H) of a graph H is a homomorphism, if it maps every edge to an edge. (Ashkara an edge; mapping two adjacent vertices to the same vertex is forbidden.)

Studying graph homomorphisms leads to a lot of very nice combinatorics and is related to logic, algebraic topology, and probability. Let Hom(G,H) be the set of homomorphisms from G to H and let t(G,H) be the probability that a random function from V(H) to V(G) is an homomorphism.  

We say that a sequence of graphs $latex (G_n)$ tends to a limit if for every fixed graph F the series $latex t(F,G_n)$  is convergent. (This concept is related to quasirandomness of graphs, to Szemeredi's regularity lemma and its recent extensions, to property testing, and to various other notions.)  The limit objects for graphs are measurable symmetric functions from $latex I^2$ into $latex I$ ($latex I$ =[0,1]) and they are called "graphons". (The definition of converging sequences for hypergraphs is similar but the limit objects are more complicated.)

I did not find Balasz' nice lecture online but here is a previous lecture by Lovasz. Lovasz and Sos proved that every "step function" is "finitely forcible". (I will not explain these terms here.) The new development is that the conjecture that every "finitely forcible" graphon is a "step function" was disproved and this makes the theory even richer and more interesting.

We can ask a question related to both topics.

Problem: Consider all of Kostochka's examples for Turan's problem. (Those include Turan's example as well as Brown's.) Do they have a unique limit point?

Related posts: "Extremal Combinatorics I," and over Terry Tao's blog: (Luca Trevisan) Checking the Quasirandomness of Graphs and Hypergraphs  

Arabic words which are part of Hebrew slang: ashkara - for real; sababa - cool, wonderful; walla - true;  ahla -  great; yalla - hurry up, c'mon, let's go.

The problems 

1. The $latex 3^d$ conjecture

A centrally symmetric d-polytope has at least $latex 3^d$ non empty faces.

2. The cube-simplex conjecture

For every k there is f(k) so that every d-polytope with $latex d \ge f(k)$ has a k-dimensional face which is either a simplex or combinatorially isomorphic to a k-dimsnional cube.

3. Barany's question

For every d-dimensional polytope P and every k, $latex 0 \le k \le d-1$,  is it true that $latex f_k(P) \ge \min (f_0(P),f_{d-1}(P))$?

(In words: the number of k-dimensional faces of P is at least the minimum between the number of vertices of P and the number of facets of P. )

4.  Fat 4-polytopes

For 4-polytopes P, is the quantity $latex (f_1(P)+f_2(P))/(f_0(P)+f_3(P))$ bounded from above by some absolute constant? 

5.  five-simplicial five-simple polytopes

Are there 5-simplicial 5-simple 10-polytopes? Or at least 5-simplicial 5-simple d-polytope for some d?

(A polytope P is k-simplicial if all its faces of dimension at most k, are simplices. A polytope P is k-simple if its dual P* is k-simplicial.)

And on a personal note: My beloved, beautiful,  and troubled country celebrates 60 today: happy birthday! 

Update (May 12): David Eppstein raised in a followup a sort of a dual question to Barany's. For a d-polytope with n vertices and n facets what is the maximal number of k-faces. For a fixed d and large n the free join of pentagons is conjectured to give asymptotically the best upper bound.

Update (July 29) Gunter Ziegler reminded me of the following additional problem of Barany: Is the number of saturated chains in a d-polytope bounded by some constant (depending on d) times the total number of faces (of all dimensions) of the polytope. A saturated flag is a 0-face inside a 1-face inside a 2-face ... inside a (d-1)-face. 

Background

A convex polytope is the convex hull of a finite set of points in an Euclidean space. A proper faceof a polytope P is the intersection of P with a supporting hyperplane. The empty face and P itself are regarded as trivial faces. The convex hull of a set of points which affinely span a d-dimensional space is a d-dimensional polytope or briefly a  d-polytope. Faces of polytopes are themselves polytopes, a k-face is a short way to say k-dimensional face.  0-faces are called vertices, 1-faces are called edgesand (d-1)-faces of a d-polytope are called facets. The set of faces of a polytope is a POSET (=partially ordered set) which is a "lattice",  "atomic", and "graded".  For a polytope P $latex f_k(P)$ denotes the number of k-faces of P. The f-vectorof P is the vector $latex (f_{-1}(P),f_0(P),f_2(P), \cdots , f_d(P))$. Two d-polytopes P and Q are combinatorially equivalent if there is a bijection between the faces of P and the faces of Q which preserves the order relation. 

The simplest d-polytope is the simplex: the convex hull of d+1 affinely independent points. The face lattice (=POSET of faces) of a simplex is just the Boolean lattice. A d-polytope is simplicialif all its proper faces are simplicies. A polytope is simpleif its dual is simplicial or, equivalently, if every vertex is included in precisely d edges. A d-polytope is k-simplicial if all its k-faces are simplices. (Here d>k.) The d-cube is the convex hull of all vextors of length d with entries +1/-1.

The five open questions in this post belong to the combinatorial theory of polytopes that deal with the set of faces of polytopes. (There are also interesting metrical and arithmetic questions about polytopes.)

If P is a convex polytope which contains the origin in its interior, the polar dualof P is the set of all points y so that $latex <x,y> \le 1$ for every $latex x \in P$. There is a 1-1 order reversing bijection between the faces of P and the faces of its polar.  

A polytope P is centrally symmetric if whenever x belongs to P so is -x.

Remarks:

1. The conjecture was motivated by results by Barany and Lovasz and by Stanley, who gave stronger statements for simplicial centrally symmetric polytopes, and by a theorem of Figiel, Lindenstrauss and Milman that asserts that for arbitrary CS d-polytopes that $latex \log f_0(P) \cdot \log f_{d-1}(P) \ge \gamma d$ for some absolute constant $latex \gamma >0$.  Raman Sanyal, Axel Werner, and Günter Ziegler provedthe $latex 3^d$ conjecture for d=4 and refuted some stronger conjectures I had made. All Hanner d-polytopes, namely d-polytopes obtained from [-1,1] by repeating the operations of 1) Cartesian product, 2) taking the polar, have precisely $latex 3^d$ non empty faces.

2.  The 120-cell (a 4-polytope whose facets are all dodecahedra) shows that f(2)>4, and I provedthat f(2)=5. The conjecture is not known even for simple polytopes. A weaker conjecture asserts that there is a finite list of k-polytopes that for every $latex d \ge g(d)$ every d-polytope has a k-face from the list. The fact that g(2)=3 (with the list being :{triangle, quandrangle, pentagon}) follows from Euler's theorem and perhaps even goes back to Descartes. Kleinschmidt, Meisinger and myself proved g(3)< 10. If we restrict ourselves to simple polytopes then it is true that the analogous g(k) is finite.

3. A positive answer to Barany's problem would follow from very strong (conjectural) versions of the upper bound theorem. This is poor excuse for our inability to answer this question.

4. This is a crucial problem in understanding all linear inequalities among face numbers of 4-polytopes. See, e.g. this paper by Ziegler and this by Eppstein, Kuperberg, and Ziegler.

5. A 4-simple 4-simplicial 8 polytope is known: This is the Gosset polytope.

More

Much more material on polytopes can be found in Günter Ziegler's home page and in his papers, book and in the new edition of Grunbaum's book "Convex polytopes", which he recently edited with Kaibel and Klee.

 Günter Ziegler

 Paul Erdös

1. Three warm up problems 

Here is how we move very quickly from very easy problems to very hard problems with a similar flavour.

Problem I: Let  N = {1,2, ... , n } . What is the largest size of a family $latex \cal F$  of subsets of $latex N$ such that every two sets in $latex \cal F$ have non empty intersection? (Such a family is called intersecting.) 

Answer I: The maximum is $latex 2^{n-1}$. You can achieve it by taking all subsets containing the element '1'. You cannot achieve more because from every pair of a set and its complement, you may choose only one set to the family. 

Problem II:  What is the largest size of a family $latex \cal F$  of subsets of $latex N$ such that every two sets in $latex \cal F$ their union is not $latex N$.

You can protest and claim that Problem II is just the same as Problem I. Just move to complements. Or just use the same answer. OK, lets have another Problem II, then. 

New Problem II: What is the largest size of a family $latex \cal F$  of subsets of $latex N$ such that for every two sets $latex S,R$ in $latex \cal F$ their intersection is non empty and their union is not $latex N$.

An example of such a family is the set of all sets containing the element '1' and missing the element '2'. This family has $latex 2^{n-2}$ sets. It took several years from the time this problem was posed by Erdös untill Kleitman showed that there is no larger family with this property.  

 Problem III (Erdös-Sos Conjecture) Let $latex \cal F$ be a family of graphs with N as the set of vertices. Suppose that every two graphs in the family have a triangle in common. How large can $latex \cal F$ be?

Now, the total number of graphs on n vertices is $latex 2 ^{{{n} \choose {2}}}$. (Note: we count labelled graphs and not isomorphism types of graphs.) A simple example of a family of graphs with the required property is all the graphs containing a fixed triangle. Say all graphs containing the edges {1,2},{1,3},{2,3}. This family contains 1/8 of all of graphs. Is there any larger family of graphs with the required property? Erdös and Sos conjectured that the answer is no - you cannot get a larger family. This conjecture is still open.

Vera Sos

 2. Two basic theorems about families with prescribed intersections. 

Erdös-Ko-Rado Theorem: An intersecting of k-subsets of $latex N$, when $latex 2k \le n$ contains at most $latex {{n-1} \choose {k-1}}$ sets.  

Fisher; deBruijn-Erdös Thorem: A family of subsets of $latex N$ so that every two (different) sets in the family have precisely  a single element in common has cardinality at most n.

(Erdös and deBruijn concluded that n non colinear points in the plane determine at least n line. Try to deduce it!)

All k-subsets of N containing the element '1' is an example of equality for the Erdos-Ko-Rado theorem. For the Erdös deBruijn take the family {{1} , {1,2} {1,3} ... {1,n}} or replace the first set by {2,3,...,n}, or take a finite projective plane.  

Fano plane the finite projective place of order 2.

3. The linear algebra proof of deBruijn Erdös Theorem.

The linear algebra proof of the Fisher; de Bruijn-Erdös Theorem  goes roughly as follows: Suppose that there are m sets in the family $latex A_1,A_2,\dots, A_m$. Consider the incidence matrix of the family:  The (i,j)-entry in this matrix is '1' if i belongs to $latex A_j$.

The crucial fact is that the columns $latex c_1,c_2,\dots, c_m$ of the incidence matrix are linearly independent. This gives $latex m \le n$.

How do we go about proving that the columns are linearly independent? We first assume that all sets have cardinality at least 2. Then we write $latex s = \sum \alpha_i c_i$, and compute the inner product

 $latex <s,s>=\sum \sum \alpha_i \alpha _j <c_i , c_j>$.

We note that if i and j are distinct $latex <c_i,c_j>=1$ and that $latex <c_i,c_i>=|A_i|$.

And write $latex <s,s>= \sum_{i=1}^m \alpha_i^2 (|A_i|-1) + (\sum_{i=1}^m \alpha_i)^2$.

This can vanish only if all coefficients $latex \alpha_i$ are equal to 0.  

Update: This proof is an example of "dimension arguments in combinatorics". For more examples and a general discussion see this post  in Gowers's weblog.

4. Sperner's theorem

(I wanted to indicate how Erdös-Ko-Rado theorem is proved. There are various proofs and for the two proofs I like to give it is better to demonstrate the proof technique in a simple case.)

Sperner's theorem fron 1927 asserts that the maximum size of a family $latex \cal F$ of subsets of N which is an antichain with respect to inclusion is the middle binomial coefficient $latex {{n} \choose {n/2}}$.  Lubell found a simple nice proof for Sperner's theorem: Let $latex \cal F$ be such an antichain and suppose that it has $latex s_k$ sets of cardinality k.  Count pairs $latex (\pi, S)$ where $latex \pi $= $latex ((\pi(1),\pi(2), \dots , \pi(n))$ is a permutation of {1,2, ... ,n} and S is a set in the family which is initial w.r.t. $latex \pi$, namely S={$latex \pi(1), \pi(2),\dots,\pi(k)$ }  for some k. Now, for every permutation $latex \pi$ you can find at most one initial S in the family $latex \cal F$ (because of the untichain condition). If S is a set of k elements, you can find precisely k! (n-k)! permutations $latex \pi$ for which S is initial. Putting these two facts together we get that $latex \sum_{k=0}^n s_k k! (n-k)! \le n!$ or in other words $latex \sum_{k=0}^n s_k/ {{n} \choose {k}} \le1$. This inequality (called the LYM inequality) implies the required result.

Bella Bollobas, one of the discoverers of the LYM inequality.

There is a similar proof for Erdös-Ko-Rado's theorem

The idea is to count pairs $latex (\pi,S)$ where $latex S$ is a set in the family, $latex \pi$ is a circular permutation $latex (\pi(1),\pi(2), \dots , \pi(n))$ and $latex S$ is a continuous "interval" with respect tp $latex \pi$.

On the one hand there are (n-1)! cyclical permutations and as it is not hard to see for each such permutation, you can get at most k "intervals" which are pairwise intersecting. On the other hand, for every set S there are k!(n-k)! cyclic permutations on which S is a continuous interval.

So $latex |\cal F|$ k! (n-k)! $latex \le (n-1)! k$ and this gives the Erdös Ko Rado theorem.

What about the "not hard to see part". This uses the fact that $latex 2k \le n$. One way about it is to consider the interval J whose left most element z is furthest to the left and notice that there are k intervals that intersect J whose left most element is right of z. Another way is to consider some interval J of length k and notice that the 2k-2 intervals intersecting it come in (k-1) pairs where each pair contains two disjoint intervals.   

5. Turan's theorem and Turan's problem 

The special case of Turan's theorem for graphs with no triangles was proved by Mantel in 1907

The maximum number $latex t_2(n)$ of edges in a graph on n vertices without a triangle is attained by a complete multi-partite graph with n vertices where the sizes of the parts are as equal as possible.

The full Turan's theorem was proved in the 40s.

The maximum number $latex t_r(n)$ of edges in a graph on n vertices which does not contain a complete subgraph with (r+1) vertices is attained by a complete multi-partite graph with n vertices where the sizes of the parts are as equal as possible.

Paul Turan

Proving Turan's theorem: It is difficult not to prove Turan's theorem; it seems that every approach to proving it succeeds. One approach is this: for simplicity consider the case of triangles. Take a vertex with maximum degree and divide the other vertices of the graph into two parts: A - the neighbors of v, B - the remaining vertices. Now, note that the vertices in A form an independent set (i.e. there are no edges between vertices in A). For every vertex in B delete all edges containing it and instead, connect it to all edges in A. Note that in the new graph, the degree of every vertex is at least as large as in the original graph. And, in addition, the new graph is bipartite. (One part is A.) It is left to show that for a bipartite graph the number of egdes is maximum, when the parts are as equal as possible.

Here is another proof. Delete a vertex from a graph G with n+1 vertices without $latex K_r$. The number of edges in the remaining graph is at most $latex t_r(n)$. Do it for all vertices and note that every edge is counted n-1 times. You get that the number of edges in G (hence $latex t_r(n+1)$) is at least the upper integral part of $latex t_r(n) \cdot (n+1)/(n-1)$. Lo and behold this gives the right formula.

So let us conclude with Turan's 1940 problem. You want to know what is the maximum cardinality of a set of triples from {1,2,...,n} which does not contain a "tetrahedron", namely four triples of the form {a,b,c},{a,b,d),{a,c,d},{b,c,d}.

If you do not know it already; try to guess and suggest the best example. Turan made a conjecture which is still open.

(May 4, a few typos corrected, thanks alef and mike.)

My name is Gil Kalai and I am a mathematician working mainly in the field of Combinatorics.  Within combinatorics, I work mainly on geometric combinatorics and the study of convex polytopes and related objects, and on the analysis of Boolean functions and related matters. I am a professor at the Institute of Mathematics at the Hebrew University of Jerusalem and also have a  long-term visiting position at the departments of Computer Science and Mathematics at Yale University, New Haven.  

 Gosset polytope- a hand drawing by Peter McMullen of the plane projection of the 8-dimensional 4-simplicial 4-simple Gosset polytope.

The Shlegel diagram (a certain 3-d projection) of the 120-cell -  a regular 4-dimensional polytope with 120 dodecahedral facets.  

2. Getting started:

I have decided to follow the (separate) suggestions of Terry Tao, Noam Nisan and Oded Schramm (followed by a final push by Avi Wigderson) and set up a blog; I plan to do it for about a year.  It will be centered around combinatorics which is my main area of research and will touch on other matters.

Tao, Schramm and Nisan

Wigderson (left)

3. A problem: Frankl's Conjecture

Let me tell you about a problem in extremal combinatorics that I like: I will probably return to the context later. But meanwile have a look:

 Frankl's conjecture

Let $latex \cal F$ be a finite family of finite sets which is closed under union. In other words,  if $latex S,T \in {\cal F}$ then also $latex S \cup T \in {\cal F}$.

There exists an element $latex x$ which belongs to at least half the sets in $latex \cal F$.

Peter Frankl

4. Earlier blog experience:

I became acquainted with blogs quite recently; Greg Kuperberg told me about Dave Bacon's blog "The Quantum Pontiff". Then I asked my children if they knew what blogs were, and they looked at me (again) like I said something very silly;  It turned out my youngest son, Lior,  had already completed a blog.  

Greg Kuperberg (left) with his mother Krystyna.  (Both Greg's parents, Krystyna and Wlodek, are well known mathematicians). Dave Bacon (right) 

Since then, I took part in a few blog discussions. I am curious about the meaning and role of mathematical discussions in general, and on blogs in particular.

5. Are mathematical debates possible?

I even took part in little "mathematical debates". Here is a little quote from one of them; never mind the context. There aren't many debates and controversies in mathematics, so if mathematical debating will ever become more developed, mathematical polemics may, perhaps, look like this:

Comment by Gil Kalai 3/14/2006:

 In all these cases you approximate a rank-one matrix to start with. I believe that you may be able to approximate a rank-one matrix up to a rank-one error. I do not believe that you will be able to approximate an arbitrary matrix up to a rank one matrix.

Comment by Dave Bacon, later that day:

I will never look at rank one matrices the same ;)

6. Guest Blogging on "what's new":

 I gave two guest posts over Terry Tao's blog

One was about the  weak epsilon net problem.

and the other was about the entropy/influence conjecture.

7.  Other topics.

I will occasionally try to discuss areas of mathematics that touch on combinatorics, and also topics that go beyond my expertise: applied mathematics, and in particular its applications to, and connection with computer science, economics, statistics,  and even physics and philosophy;  and issues related to academic life, especially in Israel.

Basically, the "toy problem" can be stated as follows:

Problem (Bochi and Fayad, 2006). Let $latex A_0\in SL(2,\mathbb{R})$ be an arbitrary hyperbolic matrix. Is it true that there are some constants $latex 0<\theta<1$ and $latex \gamma>1$ such that for almost every matrix $latex A_1\in SL(2,\mathbb{R})$ and for every word $latex w\in\{0,1\}^{\mathbb{N}}$ satisfying the frequency condition :

(1) $latex \#\{j\in\{1,\dots,N\}: w_j=1\}\leq \theta N,$

we have

(2) $latex \|A_{w_N}\dots A_{w_1}\|>\gamma^N$

for all $latex N\in\mathbb{N}$?

Before proceeding further, let me explain the italic terms in the statement of the problem and what makes this question very intuitive and natural.

First of all, I recall the concept of hyperbolic matrix. The usual way to define hyperbolicity is the following: we say that an invertible $latex n\times n$ matrix A is hyperbolic if and only if none of its eigenvalues lies on the unit circle $latex S^1$. In other words, A is hyperbolic whenever the modulus of all of its eigenvalues is different from 1. Of course, hyperbolic $latex n\times n$ matrices are pleasant objects since they admit a simple description of its dynamics: we can decompose $latex \mathbb{R}^n = E\oplus F$ into the sum of two generalized eigenspaces (E being the sum of the eigenspaces of eigenvalues with modulus less than 1 and F being the sum of eigenspaces of eigenvalues with modulus greater than 1) so that the action of A restricted to these two subspace is very simple - $latex A|_E$ is a contraction and $latex A|_F$ is an expansion.

It turns out that when dealing with $latex SL(2,\mathbb{R})$ matrices (i.e., $latex 2\times 2$ invertible matrices with determinant 1), we can define hyperbolicity in a short manner: $latex A\in SL(2,\mathbb{R})$ is hyperbolic if and only if $latex |tr A|>2$ (where $latex tr A$ stands for the trace of A). It is an easy exercise of linear algebra left to the reader to check that these two concepts of "hyperbolicity" for $latex SL(2,\mathbb{R})$ matrices are equivalent.

Next, let me explain the meaning of "almost every matrix of $latex SL(2,\mathbb{R})$". It is well-known that $latex SL(2,\mathbb{R})$ is a three-dimensional manifold (I think that the best way to see this is using the polar decomposition theorem from linear algebra. It implies that any matrix A of $latex SL(2,\mathbb{R})$ can be written as $latex A = R_{b} H_c R_{a}$ where $latex R_{\mu}$ denotes the rotation by angle $latex \mu$ and $latex H_c = diag (c, 1/c)$ is the diagonal matrix with entries $latex c, c^{-1}$ and $latex c\geq 1$. Hence, these three parameters a,b,c serves as a local coordinate system). In particular, we can speak about subsets of "zero measure" on this 3-manifold (by just looking at the zero Lebesgue measure sets on the local charts). A more advanced way to introduce the notion of zero measure subsets of $latex SL(2,\mathbb{R})$ uses its Haar measure, which is a natural measure generalizing the notion of Lebesgue measure (in the sense that this measure is "invariant by translations") to the setting of locally compact groups (such as $latex \mathbb{R}$ and $latex SL(2,\mathbb{R})$).

After these preliminaries, we are ready to discuss some reasons making Bochi and Fayad question a very natural one. We begin by noticing that any hyperbolic matrix $latex A_0\in SL(2,\mathbb{R})$ verifies $latex \|A_0^N\| = \|A_0\|^N = \lambda^N$ where $latex \lambda>|tr A_0|/2>1$ is the biggest eigenvalue of A. Hence, if our word $latex w\in\{0,1\}^{\mathbb{N}}$ possesses only 0's, it follows that the estimate (2) holds. Now consider more general words satisfying the frequency condition (1) above. To simplify our explanation, let's assume that $latex A_1\in SO(2,\mathbb{R})$ is a rotation $latex R_{\mu}$ of angle $latex \mu$. In this situation, the problem asks about the growth rate of the norm of a certain products of the matrices $latex A_0$ and $latex R_{\mu}$ for Lebesgue almost every $latex \mu$. Of course, we can't hope to control the norm of a product of these two matrices, at least when the proportion of 0's and 1's are more or less equal (i.e., when the word is balanced). In fact, if $latex A_0 = diag (3, 1/3)$ and $latex A_1=R_{\pi/2}$, then it is not hard to see that the condition (2) is not verified for the word $latex w = (1010101010101010...)$. Indeed, by direct computation, one sees immediately that $latex \|A_{w_N}\dots A_{w_1}\|\leq 3$ for all $latex N\in\mathbb{N}$. Looking more closely at this simple example, we note that there are two effects playing a role here: while the hyperbolic matrix $latex A_0$ tends to expand the norm of almost all vectors (i.e., it will expand the norm of every vector except for its stable vector $latex e_2 = (0,1)$), the rotation $latex R_{\pi/2}$ sends the expanded vectors to the contracted vectors by $latex A_0$, so that any growth gained in a previous step is completely lost in the next turn of the multiplication. In particular, in the case of a balanced word, these two effects fight one against the other but none of them wins at the end (and our products of matrices have bounded norm during this process).

On the other hand, if we impose a strong frequency condition, such as (1) with $latex \theta\ll 1$ sufficiently small (perhaps depending on the size of the biggest eigenvalue $latex \lambda$ of $latex A_0$), we can hope to get the estimate (2) by the following intuitive argument: whenever we see a long string of 0's in our word, we are multiplying several times the matrix $latex A_0$ by itself, so that we see a lot of expansion of the unstable vector of $latex A_0$ (i.e., the eigenvector associated to the biggest eigenvalue $latex\lambda$; for instance, this vector is $latex e_1=(1,0)$ for the diagonal matrix considered in the previous paragraph). Of course, since we are eventually multiplying by rotations, there is an inevitable loss of norm at certain points of the word. However, since there are only a few rotations involved here, it is reasonable to expect an exponential growth of the norm of our products of matrices except if we are sufficiently unlucky to look at a rotation which sends our best expanding vectors to the stable direction of $latex A_0$ (for instance, the stable direction of the diagonal matrix of the previous paragraph is $latex e_2=(0,1)$). But, the probability of these "bad" rotations described above should be small (as the size of our words grows) because we have just few rotations to deal with.

Unfortunately, it doesn't seem easy to formalize this naive argument. Nevertheless, we should point out that Fayad and Krikorian were able to partly solve this problem under a very restrictive frequency condition. By the way, I plan to discuss later in this blog this recent theorem of Fayad and Krikorian.

Let me close this post just by saying that the conclusion "for almost every $latex A_1\in SL(2,\mathbb{R})$" can't be improved: Bochi and Fayad proved that the set of matrices $latex A_1\in SL(2,\mathbb{R})$ such that (2) don't hold is residual (of course, this result doesn't answers the problem because it is well-known that there are residual sets of zero measure).