Off-diagonal Ramsey numbers of 3-graphs

Abstract

We prove a conjecture of Conlon, Fox, Gunby, He, Mubayi, Suk, Verstra\"{e}te, and Yu that for a $3$-graph $H$, the off-diagonal Ramsey number $R(H,n)$ is polynomial in $n$ if and only if $H$ is isomorphic to a subgraph of the"depth-$t$ iterated blowup"$\textsf{B}_t$ for an integer $t\geq 0$. Here, the $3$-graphs $\textsf{B}_t$ are defined recursively: $\textsf{B}_0$ has one vertex, and $\textsf{B}_t$ for $t\geq 1$ is obtained from the disjoint union of three copies of $\textsf{B}_{t-1}$ by adding all transversal hyperedges.