The Maker-Breaker Rado game on a random set of integers

TL;DR

This paper investigates the Maker-Breaker game on random integer sets, establishing threshold probabilities for Maker or Breaker winning, especially for matrices related to linear equations and Rado's theorem.

Contribution

It determines the threshold probability for Maker or Breaker victory in the Maker-Breaker $(A,b)$-game on random sets, extending to matrices satisfying Rado's theorem.

Findings

Identifies threshold probability $p_0$ for game outcomes.

Extends results to matrices related to linear equations.

Includes matrices satisfying Rado's partition theorem.

Abstract

Given an integer-valued matrix $A$ of dimension $\ell \times k$ and an integer-valued vector $b$ of dimension $\ell$, the Maker-Breaker $(A,b)$-game on a set of integers $X$ is the game where Maker and Breaker take turns claiming previously unclaimed integers from $X$, and Maker's aim is to obtain a solution to the system $Ax=b$, whereas Breaker's aim is to prevent this. When $X$ is a random subset of $\{1,\dots,n\}$ where each number is included with probability $p$ independently of all others, we determine the threshold probability $p_0$ for when the game is Maker or Breaker's win, for a large class of matrices and vectors. This class includes but is not limited to all pairs $(A,b)$ for which $Ax=b$ corresponds to a single linear equation. The Maker's win statement also extends to a much wider class of matrices which include those which satisfy Rado's partition theorem.