Harnessing the Bethe Free Energy

Gibbs measures induced by random factor graphs play a prominent role in computer science, combinatorics and physics. A key problem is to calculate the typical value of the partition function. According to the "replica symmetric cavity method", a heuristic that rests on non-rigorous considerations from statistical mechanics, in many cases this problem can be tackled by way of maximising a functional called the "Bethe free energy". In this paper we prove that the Bethe free energy upper-bounds the partition function in a broad class of models. Additionally, we provide a sufficient condition for this upper bound to be tight

Small union with large set of centers

Let $T\subset{\mathbb R}^n$ be a fixed set. By a scaled copy of $T$ around $x\in{\mathbb R}^n$ we mean a set of the form $x+rT$ for some $r>0$. In this survey paper we study results about the following type of problems: How small can a set be if it contains a scaled copy of $T$ around every point of a set of given size? We will consider the cases when $T$ is circle or sphere centered at the origin, Cantor set in ${\mathbb R}$, the boundary of a square centered at the origin, or more generally the $k$-skeleton ($0\le k<n$) of an $n$-dimensional cube centered at the origin or the $k$-skeleton of a more general polytope of ${\mathbb R}^n$. We also study the case when we allow not only scaled copies but also scaled and rotated copies and also the case when we allow only rotated copies

The Szemeredi-Trotter Theorem in the Complex Plane

It is shown that $n$ points and $e$ lines in the complex Euclidean plane ${\mathbb C}^2$ determine $O(n^{2/3}e^{2/3}+n+e)$ point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemer\'edi and Trotter about point-line incidences in the real Euclidean plane ${\mathbb R}^2$.Comment: 24 pages, 5 figures, to appear in Combinatoric

The critical window for the classical Ramsey-Tur\'an problem

The first application of Szemer\'edi's powerful regularity method was the following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.Comment: 34 page

Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \epsilon may require as many as 2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and \al, how large Hausdorff dimension can a compact set A\su\Rn have if $A$ does not contain three points that form an angle \al? (2) Given \al and \de, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the \de-neighborhood of \al? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^\circ$ seems to behave differently from other angles

Regularity inheritance in pseudorandom graphs

Advancing the sparse regularity method, we prove one-sided and two-sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox, and Zhao. These inheritance lemmas also imply improved H-counting lemmas for subgraphs of bijumbled graphs, for some H

A Bichromatic Incidence Bound and an Application

We prove a new, tight upper bound on the number of incidences between points and hyperplanes in Euclidean d-space. Given n points, of which k are colored red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between the k red points and m hyperplanes spanned by all n points provided that m = \Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal and Aronov. We use this incidence bound to prove that a set of n points, no more than n-k of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also provide an infinite family of counterexamples to a conjecture of Purdy's on the number of hyperplanes spanned by a set of points in dimensions higher than 3, and present new conjectures not subject to the counterexample.Comment: 12 page