Ordered Ramsey numbers of 3-uniform hypergraphs with bounded weak degeneracy

Abstract

The \emph{ordered Ramsey number} $r_{<}(G,H)$ of ordered $k$-graphs $G$ and $H$ is the least integer $N$ such that every red-blue edge-coloring of the naturally ordered complete $k$-graph on $[N]$ contains a blue ordered copy of $G$ or a red ordered copy of $H$. We prove that there is an absolute constant $c>0$ such that, for every integer $d\ge 1$, there is a constant $C_d>0$ for which every weakly $d$-degenerate ordered $3$-graph $H$ on $t$ vertices satisfies \[ r_<\bigl(H,K_3^{(3)}(n)\bigr) \le t\,2^{C_d n^{2-c/d}} \] for every positive integer $n$. This resolves a problem posed by Balko and Vizer ({\em SIAM J. Discrete Math., 2022}) in a stronger form. Furthermore, we show that the weak-degeneracy hypothesis cannot be replaced by bounded standard degeneracy. In particular, for every sufficiently large $n$, there exists a $1$-degenerate ordered $3$-graph $F$ on at most $2^{O(n)}$ vertices such that $r_{<}(F,K_3^{(3)}(n))>2^{\Omega (n^2)}.$