On Ordered Ramsey Numbers of Tripartite 3-Uniform Hypergraphs

Abstract

For $k \ge 2$, an ordered k-uniform hypergraph $\mathcal {H}=(H,<)$ is a k-uniform hypergraph H together with a fixed linear ordering < of its vertex set. The ordered Ramsey number $\overline{R}(\mathcal {H},\mathcal {G})$ of two ordered k-uniform hypergraphs $\mathcal {H}$ and $\mathcal {G}$ is the smallest such that every red-blue coloring of the hyperedges of the ordered complete k-uniform hypergraph $\mathcal {K}^{(k)}_N$ contains a blue copy of $\mathcal {H}$ or a red copy of $\mathcal {G}$.

The ordered Ramsey numbers are quite extensively studied for ordered graphs, but little is known about ordered hypergraphs of higher uniformity. We provide some of the first nontrivial estimates on ordered Ramsey numbers of ordered 3-uniform hypergraphs. In particular, we prove that for all and for every ordered 3-uniform hypergraph $\mathcal {H}$ on n vertices with maximum degree d and with interval chromatic number 3 there is an $\varepsilon =\varepsilon (d)>0$ such that $\overline{R}(\mathcal {H},\mathcal {H}) \le 2^{O(n^{2-\varepsilon })}$.