A central theme in additive combinatorics is the study of the structure of sets with small doubling, i.e. sets A such that A + A has size not much larger than A. In this talk we discuss a problem of a similar flavour for set systems. Given a set-family F, let F*F={A ∩ B | A,B ∈ F}. We measure the size of F by the dimension of the subspace spanned by the characteristic vectors of the sets in F over some field. What can we say about F if F*F is not much larger than F? Observe that if S is an atomic set-family, i.e., the ground set is split into disjoint subsets and S contains their arbitrary unions, then S* S = S. Our structure theorem shows that this is essentially the only possible example: any set-family F with small ‘doubling’ must be close to being atomic. We will use this theorem to solve a 40-year old problem of Frankl and Odlyzko on set families with restricted intersections.
Joint work with L. Gishboliner and I. Tomon
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
