Online Ramsey numbers of ordered paths and cycles

TL;DR

This paper investigates the online ordered Ramsey numbers for paths and cycles, providing bounds and classifications for various fixed graphs G, revealing how these numbers grow with the size of the ordered paths and cycles.

Contribution

It establishes new bounds on online ordered Ramsey numbers for paths and cycles, including an $O(n \, \log n)$ bound for all G and an $O(n)$ bound for 3-ichromatic G, and classifies graphs with linear growth.

Findings

Proves $O(n \log n)$ bound for all G and $O(n)$ for 3-ichromatic G.

Partially classifies graphs with $r_o(G,P_n) = n + O(1)$.

Extends results to ordered cycles $C_n$.

Abstract

An ordered graph is a graph with a linear ordering on its vertices. The online Ramsey game for ordered graphs $G$ and $H$ is played on an infinite sequence of vertices; on each turn, Builder draws an edge between two vertices, and Painter colors it red or blue. Builder tries to create a red $G$ or a blue $H$ as quickly as possible, while Painter wants the opposite. The online ordered Ramsey number $r_o(G,H)$ is the number of turns the game lasts with optimal play. In this paper, we consider the behavior of $r_o(G,P_n)$ for fixed $G$, where $P_n$ is the monotone ordered path. We prove an $O(n \log_2n)$ bound on $r_o(G,P_n)$ for all $G$ and an $O(n)$ bound when $G$ is $3$-ichromatic; we partially classify graphs $G$ with $r_o(G,P_n) = n + O(1)$. Many of these results extend to $r_o(G,C_n)$, where $C_n$ is an ordered cycle obtained from $P_n$ by adding one edge.