Off-diagonal online size Ramsey numbers for paths

Abstract

Consider the following Ramsey game played on the edge set of KN. In every round, Builder selects an edge and Painter colours it red or blue. Builder’s goal is to force Painter to create a red copy of a path Pk on k vertices or a blue copy of Pn as soon as possible. The online (size) Ramsey number r̃(Pk,Pn) is the number of rounds in the game provided Builder and Painter play optimally. We prove that r̃(Pk,Pn)≤(5/3+o(1))n provided k=o(n) and n→∞. We also show that r̃(P4,Pn)≤⌈7n/5⌉−1 for n≥10, which improves the upper bound obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo and implies their conjecture that r̃(P4,Pn)=⌈7n/5⌉−1.