Strong Ramsey game on two boards

TL;DR

This paper investigates a variant of the strong Ramsey game played on two disjoint complete graphs, proving the existence of infinitely many target graphs for which the first player cannot win in a bounded number of moves, shedding light on a longstanding open problem.

Contribution

It introduces and analyzes the game on two disjoint boards, demonstrating infinitely many graphs where the first player lacks a bounded winning strategy, advancing understanding of the open problem.

Findings

Existence of infinitely many graphs $H$ with no bounded winning strategy for $P_1$ on $K_n  K_n$

Concise proof establishing non-existence of bounded winning strategies in the variant game

Evidence supporting the possibility of graphs that challenge the longstanding open problem

Abstract

The strong Ramsey game $R(\mathcal{B}, H)$ is a two-player game played on a graph $\mathcal{B}$, referred to as the board, with a target graph $H$. In this game, two players, $P_1$ and $P_2$, alternately claim unclaimed edges of $\mathcal{B}$, starting with $P_1$. The goal is to claim a subgraph isomorphic to $H$, with the first player achieving this declared the winner. A fundamental open question, persisting for over three decades, asks whether there exists a graph $H$ such that in the game $R(K_n, H)$, $P_1$ does not have a winning strategy in a bounded number of moves as $n \to \infty$. In this paper, we shift the focus to the variant $R(K_n \sqcup K_n, H)$, introduced by David, Hartarsky, and Tiba, where the board $K_n \sqcup K_n$ consists of two disjoint copies of $K_n$. We prove that there exist infinitely many graphs $H$ such that $P_1$ cannot win in $R(K_n \sqcup K_n, H)$ within a bounded number of moves through a concise proof. This perhaps provides evidence for the existence of examples to the above longstanding open problem.