On-line size Ramsey number for monotone k-uniform ordered paths with uniform looseness

TL;DR

This paper investigates the on-line size Ramsey numbers for monotone k-uniform ordered paths, providing bounds that improve previous results and extending to paths with loose edges, advancing understanding in ordered hypergraph Ramsey theory.

Contribution

The paper establishes new bounds for on-line size Ramsey numbers of monotone paths in ordered hypergraphs, including generalizations to loose paths, improving upon prior bounds especially for large number of colors.

Findings

Derived bounds for t-color on-line size Ramsey numbers of monotone paths.

Extended results to ll-loose monotone paths.

Improved bounds over previous work when the number of colors grows faster than m/log m.

Abstract

An ordered hypergraph is a hypergraph $H$ with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph $G$ in $H$ must respect the specified order on $V(G)$. In on-line Ramsey theory, Builder iteratively presents edges that Painter must immediately color. The $t$-color on-line size Ramsey number $\tilde R_t (G)$ of an ordered hypergraph $G$ is the minimum number of edges Builder needs to play (on a large ordered set of vertices) to force Painter using $t$ colors to produce a monochromatic copy of $G$. The monotone tight path $P_r^{(k)}$ is the ordered hypergraph with $r$ vertices whose edges are all sets of $k$ consecutive vertices. We obtain good bounds on $\tilde R_t (P_r^{(k)})$. Letting $m=r-k+1$ (the number of edges in $P_r^{(k)}$), we prove $m^{t-1}/(3\sqrt t)\le\tilde R_t (P_r^{(2)})\le tm^{t+1}$. For general $k$, a trivial upper bound is ${R \choose k}$, where $R$ is the least number of vertices in a $k$-uniform (ordered) hypergraph whose $t$-colorings all contain $P_r^{(k)}$ (and is a tower of height $k-2$). We prove $R/(k\lg R)\le\tilde R_t(P_r^{(k)})\le R(\lg R)^{2+\epsilon}$, where $\epsilon$ is any positive constant and $t(m-1)$ is sufficiently large. Our upper bounds improve prior results when $t$ grows faster than $m/\log m$. We also generalize our results to $\ell$-loose monotone paths, where each successive edge begins $\ell$ vertices after the previous edge.