Strong Ramsey game on two boards

TL;DR

The paper investigates strong Ramsey games, focusing on whether $P_1$ can win in a bounded number of moves for $R(K_n,H)$. It shifts attention to the two-board variant $R(K_n\sqcup K_n,H)$ and proves that there exist infinitely many target graphs $H$, specifically $H=K_{2,t+1}(t-2)$ with $t\ge 3$, for which $P_1$ cannot force a bounded-win, establishing $L(K_n\sqcup K_n,H)$ unbounded. It also provides positive results: explicit constant-range winning strategies for $P_1$ in $R(K_n, C_\ell)$ and in $R(K_n\sqcup K_n, K_{2,3})$, via constructive play. These findings offer evidence related to Beck's conjecture and motivate identifying other $H$ with small $L(K_n,H)$ while highlighting directions for future work on upper bounds and related extremal considerations, including a bound for 2-color-critical graphs.

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.

Figures (9)

• Figure 2.1: For $t=3$, all possible graphs obtained from $K_{2,4}(1)$ by removing $2$ edges (under isomorphism) .
• Figure 2.2: The graph $K_{2,3}(2)$: key of $P_{2}$ for $t=3$
• Figure 3.1: The early strategy of $P_1$
• Figure 3.2: The graphs (1) (2) and (5)
• Figure 3.3: The graphs (3) (4) and (6)

Theorems & Definitions (22)

• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Claim 2.1
• proof : Proof of claim
• Claim 2.2
• proof : Proof of claim
• Claim 2.3
• proof : Proof of claim
• Claim 2.4