Georgia Institute of Technology College of Sciences

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.

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.

This talk will be on an algebraic proof of theSzemeredi-Trotter theorem, as given by Kaplan, Matousek and Sharir.The lecture assumes no prior knowledge of advanced algebra.

The Green-Tao theorem says that the primes contain arithmetic progressions of arbitrary length. Tao and Ziegler extended it to polynomial progressions, showing that congurations {a+P_1(d), ..., a+P_k(d)} exist in the primes, where P_1, ..., P_k are polynomials in \mathbf{Z}[x] without constant terms (thus the Green-Tao theorem corresponds to the case where all the P_i are linear). We extend this result further, showing that we can add the extra requirement that d be of the form p-1 (or p + 1) where p is prime. This is joint work with Julia Wolf.