Drawing strategies in Strong Ramsey games for 3-uniform hypergraphs

TL;DR

This paper investigates the Strong Ramsey game for 3-uniform hypergraphs, demonstrating that the second player can always draw with an infinite set of target hypergraphs, improving previous results.

Contribution

It introduces an infinite set of 3-uniform hypergraphs where the second player has a guaranteed drawing strategy, advancing understanding of strategic outcomes in these games.

Findings

Second player can always draw in the game for the hypergraphs in the set

Improves previous results by David, Hartarsky, and Tiba

Establishes new strategies for 3-uniform hypergraph games

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.