Two player game variant of the Erdos-Szekeres problem

TL;DR

This paper introduces a two-player game variant of the Erdos-Szekeres problem, analyzing strategies to avoid convex polygons and proving the game concludes within nine moves.

Contribution

It presents the first strategic analysis of a two-player game variant of the Erdos-Szekeres problem, including a winning strategy for the second player.

Findings

Second player has a winning strategy in the convex 5-gon game.

The game always ends within 9 moves.

Specific configurations determine the game's outcome.

Abstract

The classical Erdos-Szekeres theorem states that a convex $k$-gon exists in every sufficiently large point set. This problem has been well studied and finding tight asymptotic bounds is considered a challenging open problem. Several variants of the Erdos-Szekeres problem have been posed and studied in the last two decades. The well studied variants include the empty convex $k$-gon problem, convex $k$-gon with specified number of interior points and the chromatic variant. In this paper, we introduce the following two player game variant of the Erdos-Szekeres problem: Consider a two player game where each player playing in alternate turns, place points in the plane. The objective of the game is to avoid the formation of the convex k-gon among the placed points. The game ends when a convex k-gon is formed and the player who placed the last point loses the game. In our paper we show a winning strategy for the player who plays second in the convex 5-gon game and the empty convex 5-gon game by considering convex layer configurations at each step. We prove that the game always ends in the 9th step by showing that the game reaches a specific set of configurations.