Online size Ramsey number for $C_4$ and $P_6$

TL;DR

This paper determines the exact online size Ramsey number for the pair of graphs $C_4$ and $P_6$, showing that Builder can win in exactly 11 rounds when both players play optimally.

Contribution

The paper refines previous bounds and proves the exact value of the online size Ramsey number for $C_4$ and $P_6$, which was previously known only within bounds.

Findings

Established that $ ilde{r}(C_4,P_6)=11$

Improved understanding of online size Ramsey numbers for specific graph pairs

Provided detailed proof of the exact number

Abstract

In this paper we consider a game played on the edge set of the infinite clique $K_\mathbb{N}$ by two players, Builder and Painter. In each round of the game, Builder chooses an edge and Painter colors it red or blue. Builder wins when Painter creates a red copy of $G$ or a blue copy of $H$, for some fixed graphs $G$ and $H$. Builder wants to win in as few rounds as possible, and Painter wants to delay Builder for as many rounds as possible. The online size Ramsey number $\tilde{r}(G,H)$, is the minimum number of rounds within which Builder can win, assuming both players play optimally. So far it has been proven by Dybizba\'nski, Dzido and Zakrzewska that $11\leq\tilde{r}(C_4,P_6)\leq13$ \cite{Dzido}. In this paper, we refine this result and show the exact value, namely we will present the Theorem that $\tilde{r}(C_4,P_6)=11$, with the details of the proof. Keywords: graph theory, Ramsey theory, combinatorial games, online size Ramsey number