# 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.

## Key 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.