Fractional Sylvester-Gallai

Posted on January 20, 2012 by

Avi Wigderson was in town and gave a beautiful talk about an extension of Sylvester-Gallai theorem. Here is a link to the paper: Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes by Boaz Barak, Zeev Dvir, Avi Wigderson, and Amir Yehudayoff.

Sylvester-Gallai

The Sylvester-Gallai Theorem:  Let X be a finite set of n points in an eulidean space such that for every two distinct points x,y \in X the line through x and y contains a third point z \in X. Then all points in X are contained in a line. 

I heard about this result when I took Benjy Weiss’s mathematics course for high-school students in 1970/1. a  The Sylvester-Gallai theorem was the last question marked with (*) in the first week’s homework. In one of the next meetings Benjy listened carefully to our ideas on how to prove it and then explained to us why our attempts of proving it are doomed to fail: What we tried to do only relied on the very basic incidence axioms of Euclidean geometry but the Sylvester-Gallai theorem does not hold for finite projective planes. (Sylvester conjectured the result in 1893. The first proof was given by Mechior in 1940 and Gallai proved it in 1945.)

My MO question

Befor describing the new results let me mention my third ever MathOverflow question that was about potential extensions of the G-S theorem. The question was roughly this: 

Suppose that V is an r dimensional variety embedded into n space so that if the intersection of every j-dimensional subspace with V is full dimensional then this intersection  is “complicated”. Then n cannot be too large.  

I will not reproduce the full question here but only a memorable remark made by Greg Kuperberg:

If you claimed that Gil is short for Gilvester (which is a real first name although rare), then you could say that any of your results is the “Gilvester Kalai theorem”. – Greg Kuperberg Nov 24 2009 at 5:13

The result by Barak, Dvir, Wigderson and Yehudayoff

Theorem:  Let X be a finite set of n points in an eulidean space such that for every point x \in X the number of y, y\in X,y \ne x such that the line through x and y contains another point of X is at least \delta (n-1). Then

\dim (Aff(X))\le 13/\delta^2

Some remarks:

1) The proof: The first ingredient of the proof is a translation of the theorem into a question about ranks of matrices with a certain combinatorial structure. The next thing is to observe is that when the non zero entries of the matrix are 1′s the claim is simple. The second surprising ingredient of the proof is to use scaling in order to “tame” the entries of the matrix. 

2)  The context – locally correctable codes:  A q-query locally correctable (q,\delta)-code over a field F is a subspace of F^n so that, given any element \tilde y that disagrees with some y \in C in at most \delta n positions and an index i, 1 \le i \le n we can recover y_i with probability 3/4 by reading at most q coordinates of \tilde y.  The theorem stated above imply that, for two queries,  over the real numbers (and also over the complex numbers), such codes do not exist when n is large. Another context where the result is of interest is the hot area of sum product theorems and related questions in the geometry of incidences. 

3) Some open problems: Is there a more detailed structure theorem for configurations of points satisfying the condition of the theorem? Can the result be improved to \dim (Aff(X))=O(1/\delta )? Can a similar result be proved on locally correctable codes with more than two queries? This also translates into an interesting Sylvester-Gallai type question but it will require, Avi said, new ideas.

Posted in Combinatorics, Computer Science and Optimization, Geometry | Tagged , , , | 1 Comment

A Theorem About Infinite Cardinals Everybody Should Know

Posted on January 18, 2012 by

Cantor proved and we all know that for every cardinal  \kappa we have

2^{\kappa}>{\kappa}.

This is a very basic fact about cardinal arithmetic and it is nice that the proof works for finite and infinite cardinals equally well. (For the finite case it looks that Cantor’s proof is genuinly different than the ordinary proof by induction.)

Do you know some other results about the arithmetic of infinite cardinals? We know that there are many statement that are independent from ZFC the axioms of sets theory but are there some results which can be proved unconditionally?

Here is a theorem of Shelah. For simplicity we will assume that the special continuum hypothesis 2^{\aleph_0}=\aleph_1.

Theorem: \prod_{i-0}^{\infty}\aleph_i <\aleph_{\omega_4}.

Here \omega_4 is the first ordinal which corresponds to \aleph_4.

Remark: without assuming the special continuum hypothesis if 2^{\aleph_0}=\aleph_{\alpha} then the theorem asserts that \prod_{i-0}^{\infty}\aleph_{\alpha+i}<\aleph_{\alpha+\omega_4}.

Want to know more? Read Uri Avraham and Menachem Magidor chapter on Cardinal Arithmetics;

Posted in Mathematical logic and set theory | Leave a comment

Ryan O’Donnell: Analysis of Boolean Function

Posted on December 18, 2011 by

Ryan O’Donnell has begun writing a book about Fourier analysis of Boolean functions and  he serializes it on a blog entiled Analysis of Boolean Function.  New sections appear on Mondays, Wednesdays, and Fridays.

Besides covering the basic theory, Ryan intends to describe applications in theoretical computer science and other areas of mathematics, including combinatorics, probability, social choice, and geometry.

Beside being a great place to learn this interesting material, actively participating in Ryan’s blog can make you a hero! Don’t miss this opportunity.

Each chapter of Ryan’s book ends with a “highlight” illustrating the use of Boolean analysis in problems where you might not necessarily expect it. In a post over Computational Complexity Ryan described some of these highlights in order to give a flavor of the contents:

  • Testing linearity (the Blum-Luby-Rubinfeld Theorem)
  • Arrow’s Theorem from Social Choice (and Kalai’s “approximate” version)
  • The Goldreich-Levin Algorithm from cryptography
  • Constant-depth circuits (Linial-Mansour-Nisan’s work)
  • Noise sensitivity of threshold functions (Peres’s Theorem)
  • Pseudorandomness for F_2-polynomials (Viola’s Theorem)
  • NP-hardness of approximately solving linear systems (Hastad’s Theorem)
  • Randomized query complexity of monotone graph properties
  • The (almost-)Polynomial Freiman-Ruzsa Theorem (i.e., Sanders’s Theorem)
  • The Kahn-Kalai-Linial Theorem on influences
  • The Gaussian Isoperimetric Inequality (Bobkov’s proof)
  • Sharp threshold phenomena (Friedgut and Bourgain’s theorems)
  • Majority Is Stablest Theorem
  • Unique Games-hardness from SDP gaps (work of Raghavendra and others)
Posted in Combinatorics, Computer Science and Optimization | Tagged , , | 1 Comment

Alexander Chervov MO’s Question: Noteworthy-Achievements-In-And-Around-2010

Posted on December 14, 2011 by

Alexander Chervov asked over Mathoverflow about Noteworthy results in and around 2010  and some interesting results were offered in the answers. If you would like to mention additional results you can comment on them here. The only requirement is to explain what the result says and give links if possible.

Posted in Uncategorized | Tagged | 5 Comments

Cup Sets, Sunflowers, and Matrix Multiplication

Posted on December 9, 2011 by

This post follows a recent paper On sunflowers  and matrix multiplication by Noga Alon, Amir Spilka, and Christopher Umens (ASU11) which rely on an earlier paper Group-theoretic algorithms for matrix multiplication, by Henry Cohn, Robert Kleinberg, Balasz Szegedy, and Christopher Umans (CKSU05), and refers also to a paper by Don Coppersmith and Shmuel Winograd (CW90).

Three famous problems

The Erdos-Rado sunflower conjecture

The Erdos-Rado sunflower (Delta system) theorem and conjecture was already menioned in this post on extremal set theory.

A sunflower (a.k.a. Delta-system) of size r is a family of sets A_1, A_2, \dots, A_r such that every element that belongs to more than oneofthe sets belongs to all of them. A basic and simple result of Erdos and Rado asserts that

Erdos-Rado sunflower theorem: There is a function f(k,r) so that every family \cal F of k-sets with more than f(k,r) members contains a sunflower of size r.

One of the most famous open problems in extremal combinatorics is:

The Erdos-Rado conjecture: Prove that f(k,r) \le c_r^k.

Here, c_r is a constant depending on r. This is most interesting already for r=3.

Three term arithmetic progressions

The cup set problem (three terms arithmetic progressions in (Z/3Z)^n):

The cup set problem was also discussed here quite extensively. (See, e.g. this post.)

Let \Gamma=\{0,1,2\}^n. The cap set problem  asks for the maximum number of elements in a subset of \Gamma which contains no arithmetic progression of size three or, alternatively, no three vectors that sum up to 0(modulo 3). (Such a set is called a cup set.) If A is a cap set of maximum size we can ask how the function h(n)=3^n/|A| behaves. Roy Meshulam proved, using Roth’s argument, that h(n) \ge n. Edell found an example of a cap set of size 2.2^n. So h(n) \le (3/2.2)^n.  The gap is exponential.

The strong cap set conjecture: h(n) \ge (1+\epsilon)^n for some \epsilon >0.

Of course, the cap set problem is closely related to

Erdos-Turan problem (for r=3): What is the larget size r_3(n) of a subest of {1,2,…,n} without 3-term arithmetic progression?

Matrix multiplications

Let ω be the smallest real number so that there is an algorithm for multiplying  two n \times n matrices which requires O(n^\omega ) arithmetic operations.

The ω=2 conjecture: ω=2.

A very recent breakthrough by Virginia Vassilevska Williams (independently) following an earlier breakthrough by Andrew Stothers improved the Coppersmith-Winograd algorithm which gave ω =2.376, to 2.374 and 2.373 respectively. (See the discussions over Lipton’s blog (1,2), Shtetl optimized, and Computational Complexity.)

It turns out that these three conjectures are related. (The connection of matrix multiplication and the Erdos-Turan problem is fairly old, but I am not sure what an even drastic improvment of Behrends’s lower bound would say about \omega.)

Three combinatorial conjectures that imply ω=2.

Remarkably, an affarmative answer for the ω=2 conjecture would folow from each one of three combinatorial conjectures. One conjecture goes back to CW90 and two were described in CKSU05. I will not present the precise formulations in order to encourage the readers to look at the original papers. (Maybe I will add the formulations later.)

The no disjoint equivoluminous subsets conjecture (CW90).

The Strong UPS conjecture (CKSU05).

Theorem: Conjecture CW90 implies the strong UPS conjecture.

CKSU’s two-family conjecture (CKSU05).

Relations between these problems

Here are some results taken from ASU11 about the relations between these combinatorial questions. The first result goes back to Erdos and Szemeredi.

The weak sunflower conjecture: A family \cal F of subsets of {1,2,…,n}  with no sunflower of size 3 can have at most (2-\epsilon)^n sets.

The following results are not difficult.

Theorem: The strong sunflower conjecture implies the weak sunflower conjecture.

Theorem: The strong cup set conjecture also implies the weak sunflower conjecture.

Theorem: The weak sunflower conjecture implies that the CW90 conjecture is false.

It follows that CW90 conjecture is in conflict both with the Erdos Rado sunflower conjecture and with the strong cup set conjecture.

Theorem: The strong cup set conjecture implies that the strong UPS conjecture is false.

While two family theorems are quite popular in extremal combinatorics (see this post and this one), CKSU’s two family conjecture is still rather isolated from other combinatorics.

What to believe?

This is a nice topic for discussion.

Posted in Combinatorics, Computer Science and Optimization, Open problems | Tagged , , , | 1 Comment

Projections to the TSP Polytope

Posted on December 7, 2011 by

Michael Ben Or told me about the following great paper Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds by Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary and Ronald de Wolf. The paper solves an old conjecture of Yannakakis about projections of polytopes.

From the abstract: “We solve a 20-year old problem posed by M. Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the maximum cut problem and the stable set problem. These results follow from a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.”

There are many interesting aspects to this story. The starting point was a series of papers in the 80s trying to prove that P=NP by solving TSP using linear programming. The idea was to present the TSP polytope as a projection of a larger dimensional polytope described by  polynomially many linear inequalities, and solve the LP problem on that larger polytope.  Yannakakis proved that such attempts are doomed to fail, when the larger LP problem keep the symmetry of the original TSP polytope.

Yannakakis asked if the symmetry condition can be removed and this is what the new paper shows. This is a very interesting result also from the point of view of convex polytope theory.

Another exciting aspect of the paper is the use of methods from quantum communication complexity.

Posted in Computer Science and Optimization, Convex polytopes | Tagged , , , | 1 Comment

High Dimensional Expanders: Introduction I

Posted on November 12, 2011 by

Alex Lubotzky and I  are running together a year long course at HU on High Dimensional Expanders. High dimensional expanders are simplical (and more general) cell complexes which generalize expander graphs. The course is taking place in Room 110 of the mathematics building on Tuesdays 10-12. The first four hours were devoted to an introduction to the course that came with a disclaimer that some of the material is new to us as well, with a promise that the course will occasionally turn into a seminar featuring interesting speakers, and with a hope that perhaps we will be able to discover while running the course interesting questions to answer or ask. It will be too difficult for me  to follow the entire course over the blog but I would like to devote two posts to the Introduction with some remarks and backgrounds.

A brief introduction.

The introduction included the following four  parts.

Introduction to Expanders: Alex briefly described the definition of expanders, their spectral properties, the relation with random walks, the construction of expanders, the construction of Ramanujan graphs, one brief application.

Introduction to high dimensional complexes: I briefly described higher dimensional topological objects and mainly simplicial complexes, and the notions of homology and cohomology.

Introduction to high dimensional expanders: I briefly went over several possible (not equivalent) definitions: Cohomological definition; geometric and topological definitions; spectral definition;

Ramanujan graphs and complexes: Alex briefly described what they are.

In this post we will go over the first three items.  The next post will discuss the fourth item, provide credits, references and links,  and also discuss examples, connections, potential applications, and questions.

Introduction

1. Expander graphs

1.1 The definition

Let G be a graph on the vertex set V. G is an \epsilon-expander if for every set U of vertices, |U| \le |V|/2, the number of edges of G containing one vertex from U and one vertex from V \backslash U is at least \epsilon|U|.

The set of edges E_G(U,\bar U) in G from U to its complement are sometimes called the edge-boundary of U. The expansion h(G) of G is defined by

h(G)= \min\{E_G(U,\bar U): U \subset V, U \ne \emptyset, |U| \le |V|/2\}.

1.2 Spectral relation

We will mainly be interested in the case where G is a k-regular graph with n vertices and k is a fixed integer. Let A be the adjecency matrix of G. The largest  eigenvalue of A is k and let \lambda=\lambda_2 be the second largest eigenvalue. 

Theorem (Alon-Boppana): \lambda\ge 2\sqrt {k-1}-o_n(1)

Theorem (Tanner, Alon-Milman): h(G)\ge (k-\lambda)/2

Theorem (Alon): h(G) \le \sqrt {2k(k-\lambda)}

1.3 Random walks

For regular graphs, the expansion property (through the spectral interpretation) implies that (unless the graph is bipartite) the simple random walk on the graph G converges quickly to the uniform distribution on the vertices.

We will come back to examples of expander graph and to a certain application after defining high dimensional expansion.

2. High dimensional complexes and homology

We defined abstract simplicial complexes and geometric simplicial complexes K. Then we defined the chain groups C_i(K): i\ge -1 with coefficients in Z/2Z and the cochain groups C^i(K) and we identified between them. Thus an element x in the ith chain or cochain groups can be regarded as a linear combination of i-dimensional faces of K. The support of x, supp(x) is the set of i-faces of K with nonzero coefficient in x.

We described the boundary and coboundary operations.  Briefly, if x is the i chain that correspond to an i-face S, then \partial (x), the boundary operation,  is the sum of all (i-1) faces of S. This definition is extended by linearity to every i-chain. When we think about x above as an element in the ith cochain group then \delta(x) is the sum (modulo 2) of all (i+1)-faces of K containing S.  

Then we defined the vector spaces – Z^i(K) of i-cocycles, B^i(K) of i-coboundaries and the ith cohomology H^i(K)=Z^i(K)/B^i(K). We also defined boundaries, cycles and homology.  

Everything we said apply to homology with other fields of coefficients except that it was easier to define the boundary/coboundary operations over Z/2Z (for other fields of coefficients we need to worry about signs). We always consider reduced homology/cohomology without saying it.

3. High dimensional expanders

We mention several different (nonequivalent) notions of high dimensional expansions. 

3.1 (co)Homological definition:

Let K be a simplicial complex, define

h_i(K)=\min_y \max _x~\{~{\frac{|supp(y)|}{|supp(x)|}}: y \in B^{i+1}(K),~x\in (C^i(K)\backslash B^i(K)), \delta (x)=y\}

Another way to write the same thing is this:

For x\in C^i(K)\backslash B^i(K) write

h(x)=\min\{|supp(\delta x)|/(|supp(x+\delta z)|: z \in C^{i-1}(K)\}.

Then h_i(K)=\min\{~h(x): x\in C^i(K)\backslash B^i(K)\}.

If K is a d-dimensional complex we will denote h(K)=h_{d-1}(K).

Some remarks:

1) For graphs: When K is a 1-dimensional complex, the cohomological definition of h_0(K) coincides with the combinatorial definition. Every x \in C^{0}(K) is the sum of vertices in a set U \subset V of vertices. \delta (x) is the sum of edges in E(U,\bar U). The formula reduces to h(G)=\min |E(U,\bar U)/(\min |U|, |V|-|U|) over all nonempty proper subsets U of V.

2) h_i(K)=0 if and only if H_i(K)\ne 0.

3.2 Spectral definition

In this little sections we move to chain and cochain groups with real coefficients. Define the Laplacian L_i(K): C^i(K)\to C^i(K) by

L_i = \delta \partial +\partial\delta.

Let \lambda^i(K) to be the minimal eigenvalue of L_i(K).

We will say that a d-dimensional space is an i-expander in the spectral sense, if \lambda^{i}(K) is large. As before we are mainly interested in the case where i=\dim (K)-1. 

3.3 Geometric and topological definitions.

Let K be a d-dimensional simplicial complex. Let \phi be an embedding of the vertices of K to R^d. Given a d-dimensional face F of K we denote by \phi(F) the convex hull of the images under \phi of the vertices of $F$.  (Alternatively we can think about K as a geometric simplicial complex and about \phi as a mao to R^d which is an affine map on the faces of K.)

The overlap number w.r.t. \phi and a point x \in R^d, denoted by overlap(K,\phi,x) is the number of d dimensional faces in K whose image under \phi contains x. The overlap number of K is defined as

overlap(K) = \min_{\phi} \max_{x \in R^d} overlap (K,\phi,x).

The topological overlap number of K is defined the same except that you allow \phi to be a continuous function from K (regarded as a geometric simplicial complex) into R^d.

Note that for graphs, large expansion implies large overlap number.

Next post: Examples, relations between the various definitions, basic research questions. Ramanujan graphs and complexes. Applications: error correcting codes, qantum codes.  More remarks, credits and and links to relevant papers.

Posted in Combinatorics, Teaching | 4 Comments

Noise Sensitivity and Percolation. Lecture Notes by Christophe Garban and Jeff Steif

Posted on November 6, 2011 by

Lectures on noise sensitivity and percolation is a new beautiful monograph by Christophe Garban and Jeff Steif.

(Some related posts on this blog: 1, 2, 3, 4, 5)

Posted in Combinatorics, Probability | Tagged , , , , | Leave a comment

The Internet, Journals and all that.

Posted on November 3, 2011 by

Tim Gowers wrote an interesting post where he proposed in surprising many details an Internet mechanism (mixing ingredients from the arXive, blogs, MathOverflow and polymath projects) to replace Journals. Noam Nisan (who advocated similar changes over the years) wrote an interesting related post entitled the problems with Journals. 

A subsequent post by Gowers proposed a much less radical proposal. And Noam also wrote an interesting subsequent post entitled the good things about Journal.

My favorite post on this issue from Izabella Laba’s blog The accidental mathematician is entitled Random thoughts on publishing and the internet .

Posted in Academics, Mathematics over the Internet | 4 Comments

A Paradoxical Self-Referential Statement

Posted on October 29, 2011 by

A small discussion in a meeting about two decades ago.

Lior: Some people in the department think that they are wiser than what they really are

John: I am really wiser than what I think I am.

John’s statement is paradoxical (and funny). It looks similar to famous paradoxical self referential statements but it has some twist.

Posted in Philosophy | Tagged , | 1 Comment