Chasing the Threshold Bias of the 3-AP Game

Abstract

In a Maker-Breaker game there are two players, Maker and Breaker, where Maker wins if they create a specified structure while Breaker wins if they prevent Maker from winning indefinitely. A $3$-term arithmetic progression, or $3$-AP, is a sequence of three distinct integers $a,b,c$ such that $b-a=c-b$. The $3$-AP game is a biased Maker-Breaker game played on $[n]$ where every round Breaker selects $q$ unclaimed integers for every Maker's one integer. Maker is trying to select points such that they have a $3$-AP and Breaker is trying to prevent this. The main question of interest is determining the threshold bias $q^{\ast }(n)$, that is the minimum value of $q=q(n)$ for which Breaker has a winning strategy. Kusch, Rué, Spiegel and Szabó initially asked this question and proved $\sqrt{n/12-1/6}\le q^{\ast }(n)\le \sqrt{3n}$. We find new strategies for both Maker and Breaker which improve the existing bounds to \[ (1+o(1))\sqrt{\frac{n}{5.6}} \leq q^*(n) \leq \sqrt{2n} +O(1). \]