On an Erd\H{o}s-Szekeres Game

TL;DR

This paper analyzes a two-player permutation game inspired by the Erd ext{"o}s-Szekeres Theorem, determining winners and strategies for specific parameter ranges, thus extending combinatorial game theory insights.

Contribution

It introduces a new permutation game based on Erd ext{"o}s-Szekeres, providing explicit winning strategies for cases where a  b and b in {2,3,4,5}.

Findings

Identifies the winner for the game when a  b and b in {2,3,4,5}.

Provides explicit winning strategies for these cases.

Extends combinatorial game theory related to permutation patterns.

Abstract

We consider a 2-player permutation game inspired by the celebrated Erd\H{o}s-Szekeres Theorem. The game depends on two positive integer parameters $a$ and $b$ and we determine the winner and give a winning strategy when $a \geq b$ and $b \in \left\{2,3,4,5\right\}$.