Richard P. Stanley Seminar in Combinatorics: Off-diagonal hypergraph Ramsey numbers

Let $r(H,n)$ denote the minimum $N$ such that any $3$-uniform hypergraph on $N$ vertices contains either a copy of $H$ or an independent set of size $n$. A tantalizing conjecture in the area states that $r(H,n)$ is polynomial in $n$ if and only if $H$ lies in the iterated blowup of a single edge. We present recent progress towards this conjecture, including a proof of the conjecture when $H$ has at most $5$ tightly connected components. Joint work with David Conlon, Jiaxi Nie, and Logan Post.

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