Combinatorics

Given a set system S_1,...,S_n over the ground set [n], the minimum discrepancy problem seeks to find a 2-coloring of the elements of the ground set so that the discrepancy i.e. the maximum (over S_1,..., S_n) imbalance of colors in a set is minimized. In a breakthrough work, Bansal provided a…

The Ramsey number r(s,t) denotes the minimum N such that in any red-blue coloring of the edges of the complete graph on N vertices, there exists a red complete graph on s vertices or a blue complete graph on t vertices. While the study of these quantities goes back almost one hundred…

A system of linear equations is Sidorenko over F_p if any subset of F_p^n contains at least as many solutions to it as a random set of the same density, asymptotically as n->infty. A system of linear equations is common over F_p if any 2-coloring of F_p^n gives at least as many monochromatic…

Consider a quadratic polynomial Q(x_1,...,x_n) of a random binary sequence (x_1,...,x_n). To what extent can Q(x_1,...,x_n) concentrate on a single value? This is a quadratic version of the classical Littlewood-Offord problem; it was was popularised by Costello, Tao and Vu in their study of…

We will describe recent progress, in joint work with Jeck Lim, on the study of sumset estimates in higher dimensions. The basic question we discuss is the following: given a subset A of d-dimensional space and a linear transformation L, how large is the sumset A + LA?

Expander graphs are perhaps one of the most widely useful classes of graphs ever considered. In this talk, we will focus on a fairly weak notion of expanders called sublinear expanders, first introduced by Komlós and Szemerédi around 30 years ago. They have found…

Let w(1),w(2),..., w(n) be positive weights. Put these weights in an urn and draw them out, without replacement, each time picking the next draw with probability proportional to its weight relative to the remaining weights. Let sigma be the resulting permutation of {1,2,...,n}. This model is…

Consider the following two-player game played on the edges of the complete graph with n vertices: In each round the first player chooses b edges, which they have not previously chosen, and the second player immediately and irrevocably picks one of them and adds it to the initially empty graph G…

A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in the 1930s that the sum of 1/(a log a), ranging over a in A, is uniformly bounded over all choices of primitive sets A. In the 1980s he asked if this sum is maximized by the set of prime…

In a vertex expanding graph, every small subset of vertices neighbors many different vertices. Random graphs are near-optimal vertex expanders; however, it has proven difficult to create families of deterministic near-optimal vertex expanders, as the connection between vertex and spectral…