The independence number of H-free hypergraphs

Series
Combinatorics Seminar
Time
Friday, February 21, 2025 - 3:15pm for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xiaoyu He – Georgia Institute of Technology – xhe399@gatech.eduhttps://alkjash.github.io/
Organizer
Xiaoyu He, Tom Kelly, Jiaxi Nie

It is a fundamental question in Ramsey theory to determine the smallest possible independence number of an H-free hypergraph on n vertices. In the case of graphs, the problem was famously solved for H=K3 by Kim and for H=K4 (up to a logarithmic factor) by Mattheus-Verstraete in 2023. Even C4 and K5 remain wide open. We study the problem for 3-uniform hypergraphs and conjecture a full classification: the minimum independence number is poly(n) if and only if H is contained in the iterated blowup of the single-edge hypergraph. We prove this conjecture for all H with at most two tightly connected components. Based on joint work with Conlon, Fox, Gunby, Mubayi, Suk, Verstraete, and Yu.