Drawing strategies in Strong Ramsey games for 3-uniform hypergraphs
Nathan Bowler, Henri Ortmüller
TL;DR
The work addresses whether the Strong Ramsey game on a countably infinite 3-uniform board admits a draw for certain finite targets. It introduces the Distraction Lemma as a pivotal technique and constructs an infinite family of 3-uniform hypergraphs $\{G_t\}$ for which the second player can force a draw in $\mathcal{R}(K_{\aleph_0}^{(3)}, G_t)$. The authors first establish a draw for $\hat{K}_{2,4}^{(3)}$ and then generalize to $K^{(3)}_{2,t+1}(t-2)$ for all $t\ge3$, thereby providing the first examples of 3-uniform hypergraphs with drawn Strong Ramsey games on a countably infinite board. These results advance understanding of draw phenomena in infinite-board Ramsey games and support related conjectures about minimal draw examples.
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. In this paper, we give an infinite set of 3-uniform hypergraphs $\{G_t\}_{t \geq 3}$, such that $P_2$ has a drawing strategy in the Strong Ramsey game $\mathcal{R}(K_{\aleph_0}^{(3)}, G_t)$. This improves a result by David, Hartarsky and Tiba.