Speaker: Marcelo Sales (UC Irvine)
Title: On possible uniform Tur\'{a}n densities.
Abstract:In the early 1980s, Erd\H{o}s and S\'{o}s introduced a variant of the classical Tur\'{a}n problem: Given a family of $3$-uniform hypergraphs $\mathcal{F}$, the \emph{uniform Tur\'{a}n density} $\pi_u(\mathcal{F})$ is defined as the infimum $d \in [0,1]$ such that every sufficiently large $3$-graph that is \emph{uniformly $d$-dense}---meaning it has edge density at least $d$ on all linearly sized subsets---contains a copy of some $F \in \mathcal{F}$. Let $\Pi_u$ denote the set of all possible uniform Tur\'{a}n densities. In this talk, we discuss recent progress on determining elements of $\Pi_u$.
Joint work with Dylan King, Sim\'{o}n Piga, Bjarne Sch\"{u}lke.