The online ordered Ramsey game is played between two players, Builder and Painter, on an infinite sequence of vertices with ordered graphs (G₁,G₂), 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 G₁ or an ordered blue copy of G₂, while Painter's objective is to delay this for as many turns as possible. The online ordered Ramsey number rₒ(G₁,G₂) 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 rₒ(G₁,G₂) in two cases: when G₁ is a cycle and G₂ a complete bipartite graph, and when G₁ is a tree and G₂ a clique.
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Heath et al. (2024) studied this question.
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