# Results on Pattern Avoidance Games

## TL;DR

This paper investigates strategies for the Pattern Avoidance Game, where two players alternately fill a zero-one matrix with ones while avoiding a specific pattern, analyzing various patterns and game generalizations.

## Contribution

It introduces new strategies for the game and explores generalizations, advancing understanding of pattern avoidance in combinatorial game settings.

## Key findings

- Identified optimal strategies for specific patterns
- Analyzed the impact of pattern complexity on game outcomes
- Extended the game to broader classes of pattern avoidance

## Abstract

A zero-one matrix $A$ contains another zero-one matrix $P$ if some submatrix of $A$ can be transformed to $P$ by changing some ones to zeros. $A$ avoids $P$ if $A$ does not contain $P$. The Pattern Avoidance Game is played by two players. Starting with an all-zero matrix, two players take turns changing zeros to ones while keeping $A$ avoiding $P$. We study the strategies of this game for some patterns $P$. We also study some generalizations of this game.