Title: “Linear dependencies, polynomial factors in the Frankl-Furedi forbidden sunflower problem”

Abstract: We call a family of s sets \{F_1, …, F_s\} a sunflower with s petals if, for any distinct i, j \in [s], one has F_i \cap F_j = \cap_{u = 1}^s F_u. The set C = \cap_{u = 1}^s F_u is called the core of the sunflower. It is a classical result of Erdos and Rado that there is a function phi(s,k) such that any family of k-element sets contains a sunflower with s petals. In 1977, Duke and Erdos asked for the size of the largest family G \subset{[n]\choose k} that contains no sunflower with s petals and core of size t-1. In 1987, Frankl and F\” uredi asymptotically solved this problem for k\ge 2t+1 and n>n_0(s,k). This paper is one of the pinnacles of the so-called Delta-system method. In this talk, we discuss the extension of the result of Frankl and F\”uredi to a much broader range of parameters:  n>f_0(s,t) k with f_0(s,t) polynomial in s and t. We also extend this result to other domains, such as [n]^k and {n\choose k/w}^w and obtain even stronger and more general results for forbidden sunflowers with core at most t-1 (including results for families of permutations and subfamilies of the k-th layer in a simplicial complex). Our methods heavily rely on properties of pseudorandom functions in the Boolean cube. In particular, we combine the spread approximation technique, introduced by Zakharov and Kupavskii, with the Delta-system approach of Frankl and F\”uredi and the hypercontractivity approach for global functions, developed by Keller, Lifshitz and coauthors. Previous works in extremal set theory relied on at most one of these methods. Creating such  a unified approach was one of the goals for our work.Joint work with Andrey Kupavskii