Date and Time: 03/03/2025, 10:30--11:30 a.m.

Location: 25 Park Place, Room 1441

Discrete Math Seminar: Off-diagonal Ramsey numbers for slowly growing hypergraphs

Speaker: Jiaxi Nie, School of Mathematics, Georgia Institute of Technology

Title: Off-diagonal Ramsey numbers for slowly growing hypergraphs

Abstract: For a $k$-uniform hypergraph $F$ and a positive integer $n$, the Ramsey number $r(F,n)$ denotes the minimum $N$ such that every $N$-vertex $F$-free $k$-uniform hypergraph contains an independent set of $n$ vertices. A  hypergraph is {\em slowly growing} if there is an ordering $e_1,e_2,\dots,e_t$ of its edges such that $|e_i \setminus \bigcup_{j = 1}^{i - 1}e_j| \leq  1$ for each $i \in \{2, \ldots, t\}$. We prove that if $k \geq 3$ is fixed and $F$ is any non $k$-partite slowly growing $k$-uniform hypergraph, then for $n\ge2$,   

\[ r(F,n) = \Omega\Bigl(\frac{n^k}{(\log n)^{2k - 2}}\Bigr).\]
In particular, we deduce that  the off-diagonal Ramsey number $r(F_5,n)$ is of order $n^{3}/\mbox{polylog}(n)$, where $F_5$ is the triple system $\{123, 124, 345\}$. This is the only 3-uniform Berge triangle for which the  polynomial power of its off-diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs,  and hypergraph containers.
This is joint with Sam Mattheus, Dhruv Mubayi and Jacques Verstraëte.

Host: Anna Yu (ajohnsen2@gsu.edu)