Geodesic
Online size Ramsey number for $C_4$ and $P_6$
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1607-1615

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In this paper we consider a game played on the edge set of the infinite clique K_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 r̃(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 Dybizbański, Dzido and Zakrzewska that 11≤r̃(C_4,P_6)≤13 [J. Dybizbański, T. Dzido and R. Zakrzewska, On-line Ramsey numbers for paths and short cycles, Discrete Appl. Math. 282 (2020) 265–270]. In this paper, we refine this result and show the exact value, namely we will present the theorem that r̃(C_4,P_6)=11, with the details of the proof.
Keywords: graph theory, Ramsey theory, combinatorial games, online size Ramsey number 
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     title = {Online size {Ramsey} number for $C_4$ and $P_6$},
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Litka, Mateusz. Online size Ramsey number for $C_4$ and $P_6$. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1607-1615. http://geodesic.mathdoc.fr/item/DMGT_2024_44_4_a19/