Off-diagonal online size Ramsey numbers for paths

Consider the following Ramsey game played on the edge set of K-N. 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 P-k on k vertices or a blue copy of P-n as soon as possible. The online (size) Ramsey number (r) over tilde (P-k, P-n) is the number of rounds in the game provided Builder and Painter play optimally. We prove that (r) over tilde (P-k, P-n) <= (5/3 + o(1))n provided k = o(n) and n -> infinity. We also show that (r) over tilde (P-4, P-n) <= [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 (r) over tilde (P-4, P-n) = [7n/5] - 1. (C) 2023 Elsevier Ltd. All rights reserved.

Automated summary

This article introduces the Ramsey game played on the edge set of K-N and investigates the online Ramsey number. The research finds that when the number of vertices k is less than n and n approaches infinity, the upper bound of the number of rounds in the game is (5/3 + o(1))n. Furthermore, it is proven that when n≥10, the upper bound of the number of rounds in the game is [7n/5] - 1, improving the previous result obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo and verifying their conjecture (r) over tilde (P-4, P-n) = [7n/5] - 1.