Ordered Ramsey numbers of loose paths and matchings

TL;DR

This paper investigates the growth of ordered Ramsey numbers for loose paths and matchings in hypergraphs, revealing their dependence on maximum degree and extending known bounds to higher uniformities.

Contribution

It determines the growth rate of ordered Ramsey numbers for loose paths and extends bounds for matchings from graphs to hypergraphs under specific orderings.

Findings

Ordered Ramsey numbers of loose paths grow as a tower of height related to maximum degree.

Extended bounds for ordered Ramsey numbers of hypergraph matchings from 2-uniform to k-uniform.

Provided upper bounds for ordered Ramsey numbers under certain orderings.

Abstract

For a $k$-uniform hypergraph $G$ with vertex set $\{1,\ldots,n\}$, the ordered Ramsey number $\operatorname{OR}_t(G)$ is the least integer $N$ such that every $t$-coloring of the edges of the complete $k$-uniform graph on vertex set $\{1,\ldots,N\}$ contains a monochromatic copy of $G$ whose vertices follow the prescribed order. Due to this added order restriction, the ordered Ramsey numbers can be much larger than the usual graph Ramsey numbers. We determine that the ordered Ramsey numbers of loose paths under a monotone order grows as a tower of height one less than the maximum degree. We also extend theorems of Conlon, Fox, Lee, and Sudakov [Ordered Ramsey numbers, arXiv:1410.5292] on the ordered Ramsey numbers of 2-uniform matchings to provide upper bounds on the ordered Ramsey number of $k$-uniform matchings under certain orderings.