Online Ramsey numbers of ordered graphs

TL;DR

This paper investigates the online ordered Ramsey number for ordered graphs, providing new lower bounds based on degrees and upper bounds for specific graph pairs, advancing understanding of this combinatorial game.

Contribution

It introduces a new degree-based lower bound and establishes upper bounds for the online ordered Ramsey number in cases involving cycles, bipartite graphs, trees, and cliques.

Findings

New lower bound based on maximum degrees.

Upper bound for cycles versus bipartite graphs.

Upper bound for trees versus cliques.

Abstract

The online ordered Ramsey game is played between two players, Builder and Painter, on an infinite sequence of vertices with ordered graphs $(G_1,G_2)$, which have linear orderings on their vertices. On each turn, Builder first selects an edge before Painter colors it red or blue. Builder's objective is to construct either an ordered red copy of $G_1$ or an ordered blue copy of $G_2$, while Painter's objective is to delay this for as many turns as possible. The online ordered Ramsey number $r_o(G_1,G_2)$ is the number of turns Builder takes to win in the case that both players play optimally. Few lower bounds are known for this quantity. In this paper, we introduce a succinct proof of a new lower bound based on the maximum left- and right-degrees in the ordered graphs. We also upper bound $r_o(G_1,G_2)$ in two cases: when $G_1$ is a cycle and $G_2$ a complete bipartite graph, and when $G_1$ is a tree and $G_2$ a clique.