$\mathcal{R}(K_{\aleph_0}, \hat{K}_{2,3})$ is a win for Player 1

TL;DR

The paper investigates whether Player 1 can force a monochromatic copy of a finite hypergraph in a Strong Ramsey game on an infinite board, focusing on the target $\\hat{K}_{2,3}$. It develops a backwards-analysis strategy based on threats, main vertices, and near-complete subgraphs to derive a winning path for $P_1$, culminating in a proof that $P_1$ wins the game $\\mathcal{R}(K_{\\aleph_0}, \\hat{K}_{2,3})$. This result advances the understanding of infinite-board Ramsey games and informs the search for draw-minimal targets, such as $\\hat{K}_{2,4}$. The techniques—threat management and backward-state reasoning—offer a structured approach for analyzing other hypergraph Ramsey games on infinite boards.

Abstract

The Strong Ramsey game $\mathcal{R}(B,G)$ is a two player game with players $P_1$ and $P_2$, where $B$ and $G$ are $k$-uniform hypergraphs for some $k \geq 2$. $G$ is always finite, while $B$ may be infinite. $P_1$ and $P_2$ alternately color uncolored edges $e \in B$ in their respective color and $P_1$ begins. Whoever completes a monochromatic copy of $G$ in their own color first, wins the game. If no one claims a monochromatic copy of $G$ in a finite number of moves, the game is declared a draw. For a $t \in \mathbb{N}$, let $\hat{K}_{2,t}$ denote the $K_{2,t}$ together with the edge connecting the two vertices in the partition class of size 2. The purpose of this paper is to give a winning strategy for $P_1$ in the game $\mathcal{R}(K_{\aleph_0}, \hat{K}_{2,3})$.

Figures (8)

• Figure 1: Parts of the board after $P_2$ blocked three times.
• Figure 2:
• Figure 3: The board after six moves by both players
• Figure 4: The critical board state after the fourth move of $P_2$.
• Figure 5:

Theorems & Definitions (10)

• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• proof : Proof of \ref{['thm:K23']}