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Series: Combinatorics Seminar

Let K^r_{r+1} denote the complete r-graph on r+1 vertices. The Turan density of K^r_{r+1} is the largest number t such that there are infinitely many K^r_{r+1}-free r-graphs with edge density t-o(1). Determining t(K^r_{r+1}) for r > 2 is a famous open problem of Turan. The best upper bound for even r, t(K^r_{r+1})\leq 1-1/r, was given by de Caen and Sidorenko. In a joint work with Linyuan Lu, we slightly improve it. For example, we show that t(K^r_{r+1})\leq 1 - 1/r - 1/(2r^3) for r=4 mod 6. One of our lemmas also leads to an exact result for hypergraphs. Given r > 2, let p be the smallest prime factor of r-1. Every r-graph on n > r(p-1) vertices such that every r+1 vertices contain 0 or r edges must be empty or a complete star.

Series: Combinatorics Seminar

Let A be a set of n real numbers. A central problem in additive combinatorics, due to Erdos and Szemeredi, is that of showing that either the sumset A+A or the product set A.A, must have close to n^2 elements. G. Elekes, in a short and brilliant paper, showed that one can give quite good bounds for this problem by invoking the Szemeredi-Trotter incidence theorem (applied to the grid (A+A) x (A.A)). Perhaps motivated by this result, J. Solymosi posed the following problem (actually, Solymosi's original problem is slightly different from the formulation I am about to give). Show that for every real c > 0, there exists 0 < d < 1, such that the following holds for all grids A x B with |A| = |B| = n sufficiently large: If one has a family of n^c lines in general position (no three meet at a point, no two parallel), at least one of them must fail to be n^(1-d)-rich -- i.e. at least one of then meets in the grid in fewer than n^(1-d) points. In this talk I will discuss a closely related result that I and Evan Borenstein have proved, and will perhaps discuss how we think we can use it to polish off this conjecture of Solymosi.

Series: Combinatorics Seminar

Let f be a polynomial or multilinear form in a large number of variables. A basic question we can ask about f is how dispersed it becomes as the number of variables increases. To be more concrete: If we randomly (and independently) set each entry to be either 1 or -1, what is the largest concentration of the output of f on any single value, assuming all (or most) of the coefficients of f are nonzero? Can we somehow describe the structure of those forms which are close to having maximal concentration? If f is a linear polynomial, this is a question originally examined by Littlewood and Offord and answered by Erdos: The maximal concentration occurs when all the nonzero coefficients of f are equal. Here we will consider the case where f is a bilinear or quadratic form.

Series: Combinatorics Seminar

This talk will detail the stability of Markov chains. One measure of stability of a time homogeneous Markov chain is a mixing time. I will define similar measures for special types of time inhomogeneous Markov chains called the adiabatic and stable adiabatic times. I will discuss the use of these Markov chains and I will discuss how the adiabatic and stable adiabatic times relate to mixing times. This talk is an exploration of linear algebra, analysis and probability.