Online size Ramsey numbers: Path vs C4

Abstract

Given two graphs G and H, a size Ramsey game is 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 G or a blue copy of H as soon as possible. The online (size) Ramsey number r˜(G,H) is the number of rounds in the game provided Builder and Painter play optimally. We prove that r˜(C4,Pn)≤2n−2 for every n≥8. The upper bound matches the lower bound obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo, so we get r˜(C4,Pn)=2n−2 for n≥8. Our proof for n≤13 is computer-assisted. The bound r˜(C4,Pn)≤2n−2 solves also the “all cycles vs. Pn” game for n≥8 – it implies that it takes Builder 2n−2 rounds to force Painter to create a blue path on n vertices or any red cycle.