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Series: Combinatorics Seminar

The problem of extending results in extremal combinatorics to sparse random and pseudorandom structures has attracted the attention of many researchers in the last decades. The breakthroughs due to several groups in the last few years have led to a better understanding of the subject, however, many questions remain unsolved. After a short introduction into this field we shall focus on some results in extremal (hyper)graph theory and additive combinatorics. Along the way some open problems will be given.

Series: Combinatorics Seminar

An open problem of interest in combinatorial number theory is that of providing a non-ergodic proof to the so called polynomial Szemeredi theorem. So far, the landmark result in this venue is that of Green who considered the emergence of 3-term arithmetic progressions whose gap is a sum of two squares (not both zero) in dense sets of integers. In view of this we consider the following problem. Given two dense subsets A and S of a finite abelian group G, what is the weakest "pseudorandomness assumption" once put on S implies that A contains a 3-term arithmetic progressions whose gap is in S? We answer this question for G=Z_n and G = F_p^n. To quantify pseudorandomness we use Gowers norms.

Series: Combinatorics Seminar

The Erdos-Szekeres (happy ending) theorem claims that among any N points in general position in the plane there are at least log_4(n) of them that are the vertices of a convex polygon. I will explain generalizations of this result that were discovered in the last 30 years involving pseudoline arrangements and families of convex bodies. After surveying some previous work I will present the following results:
1) We improve the upper bound of the analogue Ramsey function for families of disjoint and noncrossing convex bodies. In fact this follows as a corollary of the equivalence between a conjecture of Goodman and Pollack about psudoline arrangements and a conjecture of Bisztrinsky and Fejes Toth about families of disjoint convex bodies. I will say a few words about how we show this equivalence.
2) We confirm a conjecture of Pach and Toth that generalizes the previous result. More precisely we give suffcient and necesary conditions for the existence of the analogue Ramsey function in the more general case in which each pair of bodies share less than k common tangents (for every fixed k).
These results are joint work with Andreas Holmsen and Michael Dobbins.

Series: Combinatorics Seminar

Graphons are limit objects that are associated with convergent sequences of
graphs. Problems from extremal combinatorics and theoretical computer science
led to a study of graphons determined by finitely many subgraph densities,
which are referred to as finitely forcible graphons. In 2011, Lovasz and
Szegedy asked several questions about the complexity of the topological space
of so-called typical vertices of a finitely forcible graphon can be. In
particular, they conjectured that the space is always compact. We disprove the
conjecture by constructing a finitely forcible graphon such that the associated
space of typical vertices is not compact. In fact, our construction actually
provides an example of a finitely forcible graphon with the space which is even
not locally compact. This is joint work with Roman Glebov and Dan Kral.

Series: Combinatorics Seminar

A well-known problem in geometry, which may be traced back to the Renaissance artist Albrecht Durer, is concerned with cutting a convex polyhedral surface along some spanning tree of its edges so that it may be isometrically embedded, or developed without overlaps, into the plane. We show that this is always possible after an affine transformation of the surface. In particular, unfoldability of a convex polyhedron does not depend on its combinatorial structure, which settles a problem of Croft, Falconer, and Guy. Among other techniques, the proof employs a topological characterization for embeddings among immersed planar disks.

Series: Combinatorics Seminar

We study the Maker-Breaker component game, played on the edge set of a
regular graph.
Given a graph G, the s-component (1:b) game is defined as follows: in
every round Maker claims one free edge of G and Breaker claims b free
edges.
Maker wins this game if her graph contains a connected component of
size at least s; otherwise, Breaker wins the game.
For all values of Breaker's bias b, we determine whether Breaker wins
(on any d-regular graph) or Maker wins (on almost every d-regular
graph) and provide explicit winning strategies for both players.
To this end, we prove an extension of a theorem by
Gallai-Hasse-Roy-Vitaver about graph orientations without long
directed simple paths.
Joint work with Alon Naor.

Series: Combinatorics Seminar

Studying the ferromagnetic Ising model with zero applied field reduces
to sampling even
subgraphs X of G with probability proportional to
\lambda^{|E(X)|}. In this paper we present a class of Markov chains
for sampling even subgraphs, which contains the classical
single-site dynamics M_G and generalizes it to nonlocal
chains. The idea is based on the fact that even subgraphs form a
vector space over F_2
generated by a cycle basis of G. Given any cycle basis C of a
graph G, we define a Markov chain M(C) whose transitions are
defined by symmetric difference with an element of C.
We characterize cycle bases into two types: long and short.
We show that for any long cycle basis C of any graph G, M(C)
requires exponential time to mix when \lambda is small.
All fundamental cycle bases of the grid in 2 and 3 dimensions
are of this type. Moreover, on the 2-dimensional grid, short bases
appear to behave like M_G. In particular, if G has
periodic boundary conditions, all short bases yield Markov chains that
require exponential time to mix for small enough \lambda. This is
joint work with Isabel Beichl, Noah Streib, and Francis Sullivan.

Series: Combinatorics Seminar

Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property P, meaning that there is a test that, given an input function f, makes a constant number of queries to f, always accepts if f satisfies P, and rejects with positive probability if the distance between f and P is nonzero. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable. We also prove that any property that can be described as the property of decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties. [Joint work with Eldar Fischer, Hamed Hatami, Pooya Hatami, and Shachar Lovett.]

Series: Combinatorics Seminar

I will survey the major results in graph and hypergraph Ramsey theory and present some recent results on hypergraph Ramsey numbers. This includes a hypergraph generalization of the graph Ramsey number R(3,t) proved recently with Kostochka and Verstraete. If time permits some proofs will be presented.

Series: Combinatorics Seminar