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*(n), that is the minimum value of q=q(n) for which Breaker has a winning strategy. Kusch, Ru\'e, Spiegel and Szab\'o initially asked this question and proved n/12-1/6≤ q*(n)≤ 3n. We find new strategies for both Maker and Breaker which improve the existing bounds to \[ (1+o(1))n5.6 ≤ q*(n) ≤ 2n +O(1). \]