Combinatorics and Probability Seminar
Anush Tserunyan
UIUC
Independent sets in algebraic hypergraphs
Abstract: A modern trend in extremal combinatorics is extending classical results from the dense setting (e.g. Szemer\'{e}di's theorem) to the sparse random setting. More precisely, one shows that a property of a given "dense" structure is inherited by a randomly chosen "sparse" substructure. A recent breakthrough tool for proving such statements is the Balogh--Morris--Samotij and Saxton--Thomason hypergraph containers method, which bounds the number of independent sets in homogeneously dense finite hypergraphs, thus implying that a random sparse subset is not independent. Another trend in combinatorics is proving combinatorial properties for algebraic, or more generally, model theoretically definable structures. Jointly with A. Bernshteyn and M. Delcourt, we combine these trends, establishing a containers-type theorem for hypergraphs definable over an algebraically closed field: if such a hypergraph is "dense", then Zariski-generic low-dimensional sets of vertices induce a relatively "dense" subhypergraph (in particular, they are not independent).
Monday October 7, 2019 at 3:00 PM in 427 SEO