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Two player game variant of the Erdös-Szekeres problem

Abstract : The classical Erdös-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 Erdös-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 Erdös-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
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Parikshit Kolipaka, Sathish Govindarajan. Two player game variant of the Erdös-Szekeres problem. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2013, Vol. 15 no. 3 (3), pp.73--100. ⟨hal-00966382⟩

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