Online Ramsey numbers of ordered graphs
Emily Heath, Dylan King, Grace McCourt, Hannah Sheats, Justin Wisby

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 , 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 or an ordered blue copy of , while Painter's objective is to delay this for as many turns as possible. The online ordered Ramsey number 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 in two cases: when is a cycle and a complete bipartite graph, and when…
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Taxonomy
TopicsOptimization and Search Problems · Advanced Graph Theory Research · Limits and Structures in Graph Theory
