William Raynaud

Families of sets that are pairwise close

For positive integers n and d, we say that subsets A,B \subset [n] are `d-close' if the minimum cyclic distance between some element a in A and some element b in B is at most d. We say a set system F is `d-close' if every pair of sets in F are d-close, and we a say a pair of set systems F, G are `cross-d-close' if for every pair of sets A in F and B in G, A and B are d-close. We investigate the maximum possible sizes of such set systems. In particular, for each non-negative integer k, we investigate the maximum possible size of k-uniform d-close set systems (i.e., d-close set systems that contain only k-element subsets of [n]). This is a natural extension of the well-known Erdos-Ko-Rado theorem, which corresponds to the case d=0. We prove tight, stable bounds for k at most a constant fraction of n (depending on d). To do so, we employ the junta method, introduced to extremal combinatorics by Dinur and Friedgut, and significantly extended by Keller and Lifshitz. Joint work with David Ellis (Bristol).