Chasing the Threshold Bias of the 3-AP Game

TL;DR

This paper investigates the threshold bias in a biased Maker-Breaker game on integers, improving bounds for Breaker's winning strategy in preventing 3-term arithmetic progressions.

Contribution

The authors develop new strategies for both players, tightening the bounds on the threshold bias for Breaker's victory in the 3-AP game.

Findings

Lower bound improved to (1+o(1))√(n/5.6)

Upper bound improved to √(2n)+O(1)

Enhanced understanding of bias thresholds in combinatorial games

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 $\sqrt{n/12-1/6}\leq q^*(n)\leq \sqrt{3n}$. We find new strategies for both Maker and Breaker which improve the existing bounds to \[ (1+o(1))\sqrt{\frac{n}{5.6}} \leq q^*(n) \leq \sqrt{2n} +O(1). \]