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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

Series: Combinatorics Seminar

Let A be a multiplicative subgroup of Z_p^*. Define the k-fold
sumset of A to be kA={x_1+...+x_k:x_1,...,x_k in A}. Recently, Shkredov
has shown that |2A| >> |A|^(8/5-\epsilon) for |A| < p^(9/17). In this talk we
will discuss extending this result to hold for |A| < p^(5/9). In addition,
we will show that 6A contains Z_p^* for |A| > p^(33/71 +\epsilon).

Series: Combinatorics Seminar

For two graphs, G and F, we write G\longrightarrow F if every
2-coloring of the edges of G results in a monochromatic copy of F.
A graph G is k-Folkman if G\longrightarrow K_k and G\not\supset K_{k+1}. We
show that there is a constant c > 0 such that for every k \ge 2 there exists a
k-Folkman graph on at most 2^{k^{ck^2}} vertices. Our probabilistic proof is
based on a careful analysis of the growth of constants in a modified proof of the
result by Rodl and the speaker from 1995 establishing a threshold for the Ramsey
property of a binomial random graph G(n,p).
Thus, at the same time, we provide a new proof of that result (for two colors) which
avoids the use of regularity lemma.
This is joint work with Vojta Rodl and Mathias Schacht.

Series: Combinatorics Seminar

For a given finite graph G of minimum degree at least k, let
G_{p} be a random subgraph of G obtained by taking each edge
independently with probability p. We prove that (i) if p \ge
\omega/k for a function \omega=\omega(k) that tends to infinity
as k does, then G_p asymptotically almost surely contains a
cycle (and thus a path) of length at least (1-o(1))k, and (ii) if
p \ge (1+o(1))\ln k/k, then G_p asymptotically almost surely
contains a path of length at least k. Our theorems extend
classical results on paths and cycles in the binomial random graph,
obtained by taking G to be the complete graph on k+1 vertices.
Joint w/ Michael Krivelevich (Tel Aviv), Benny Sudakov (UCLA).

Series: Combinatorics Seminar

We consider higher order Markov random fields to study independent sets in
regular graphs of large girth (i.e. Bethe lattice, Cayley tree). We give
sufficient conditions for a second-order homogenous isotropic Markov
random field to exhibit long-range boundary independence (i.e. decay of
correlations, unique infinite-volume Gibbs measure), and give both
necessary and sufficient conditions when the relevant clique potentials of
the corresponding Gibbs measure satisfy a log-convexity assumption. We
gain further insight into this characterization by interpreting our model
as a multi-dimensional perturbation of the hardcore model, and (under a
convexity assumption) give a simple polyhedral characterization for those
perturbations (around the well-studied critical activity of the hardcore
model) which maintain long-range boundary independence. After identifying
several features of this polyhedron, we also characterize (again as a
polyhedral set) how one can change the occupancy probabilities through
such a perturbation. We then use linear programming to analyze the
properties of this set of attainable probabilities, showing that although
one cannot acheive denser independent sets, it is possible to optimize the
number of excluded nodes which are adjacent to no included nodes.

Series: Combinatorics Seminar

A permutation of the set {1,2,...,n} is connected if there is no k < n such
that the set of the first k numbers is invariant as a set under the
permutation. For each permutation, there is a corresponding graph whose
vertices are the letters of the permutation and whose edges correspond to
the inversions in the permutation. In this way, connected permutations
correspond to connected permutation graphs.
We find a growth process of a random permutation in which we start with the
identity permutation on a fixed set of letters and increase the number of
inversions one at a time. After the m-th step of the process, we obtain a
random permutation s(n,m) that is uniformly distributed over all
permutations of {1,2,...,n} with m inversions. We will discuss the evolution
process, the connectedness threshold for the number of inversions of s(n,m),
and the sizes of the components when m is near the threshold value. This
study fits into the wider framework of random graphs since it is analogous
to studying phase transitions in random graphs. It is a joint work with my
adviser Boris Pittel.