Online Ramsey numbers: Long versus short cycles
Grzegorz Adamski, Ma{\l}gorzata Bednarska-Bzd\c{e}ga, V\'aclav, Bla\v{z}ej

TL;DR
This paper investigates the online Ramsey numbers for cycles, showing different bounds depending on whether the cycle length is even or odd, with implications for understanding online Ramsey game dynamics.
Contribution
It provides new bounds for the online Ramsey numbers involving cycles, distinguishing between even and odd fixed cycle lengths.
Findings
For even k, r̃(C_k,C_n) = 2n + O(k)
For odd k, r̃(C_k,C_n) 3n + o(n)
Bounds depend on the parity of the fixed cycle length k.
Abstract
Online Ramsey game is played between Builder and Painter on an infinite board . In every round Builder selects an edge, then Painter colors it red or blue. Both know target graphs and . Builder aims to create either a red copy of or a blue copy of in as soon as possible, and Painter tries to prevent it. The online Ramsey number is the minimum number of rounds such that the Builder wins. We study where is fixed and is large. We show that for an absolute constant if is even, while if is odd.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Topology and Set Theory · Limits and Structures in Graph Theory
