Two player game variant of the Erdos-Szekeres problem

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.

💡 Research Summary

The paper introduces a two‑player combinatorial game that is a direct analogue of the classical Erdős‑Szekeres problem. The players alternately place points in the plane, always in general position, and the first time a convex k‑gon (or an empty convex k‑gon) appears, the player who placed the last point loses. The authors focus on the case k = 5 and define NG(k) as the minimum number of moves before the game must end, and HG(k) for the empty‑gon variant. They prove that NG(5) = HG(5) = 9 and that the second player (the player who moves on even turns) has a forced winning strategy.

The technical core of the paper is a careful analysis of the convex‑layer structure of the point set after each move. A point set P is described by its “type” (i₁,i₂,…,i_m), where i₁ is the size of the outer convex hull, i₂ the size of the second layer, and so on. The notation U(i,j) denotes a configuration with i points on the first layer and j on the second. The authors also define geometric regions called “type‑1 beam” and “type‑2 beam”, which partition the plane relative to a given convex quadrilateral into O‑regions (which would immediately create an empty convex 5‑gon) and safe I‑, Z‑, S‑regions.

The winning strategy for Player 2 proceeds as follows:

1. Turn 4 (Player 2) – Place a point so that the first four points form a parallelogram (Configuration 4). This shape is crucial because its diagonals and sides define the beam regions used later.

2. Turn 5 (Player 1) – Whatever point Player 1 chooses, it cannot lie in an O‑region (otherwise the game would already end). Hence the point falls either inside the parallelogram (Z‑region) yielding Configuration 5.1, or in one of the four external I‑regions yielding Configuration 5.2.

3. Turn 6 (Player 2) – In Configuration 5.1, Player 2 places a point symmetric to the interior point across a diagonal, creating two points that are opposite each other in the two triangles formed by the diagonals; this yields Configuration 6.1. In Configuration 5.2, Player 2 chooses a pair of opposite I‑regions (e.g., I₁ and I₈) and places a point there, producing Configuration 6.2. Both configurations keep the set free of any convex or empty 5‑gon while establishing a symmetric structure.

4. Turn 7 (Player 1) – Any safe placement again avoids O‑regions, leading either to Configuration 7.1 (the three outer‑layer points occupy three distinct I‑regions of the inner quadrilateral) or to Configuration 7.2 (four outer‑layer points with no empty 5‑gon yet).

5. Turn 8 (Player 2) – In either Configuration 7.1 or 7.2 there exists at least one “feasible region” (a portion of an I‑ or Z‑region not covered by any O‑region of the empty convex quadrilaterals formed so far). Player 2 places the eighth point there, producing Configuration 8, where each of the four outer‑layer points lies in a different I‑region of the inner quadrilateral.

6. Turn 9 (Player 1) – At this stage every possible placement of the ninth point inevitably creates a convex 5‑gon or an empty convex 5‑gon, because the eight‑point configuration already blocks all O‑regions. Consequently Player 1 loses.

The authors formalize each step with Lemmas 3.1, 3.2, and 3.3, providing exhaustive case analyses and illustrative figures (Figures 1–13). They also argue that the same reasoning applies to both the ordinary convex‑5‑gon game and its empty‑gon variant, since the feasible regions for the empty‑gon version are supersets of those for the ordinary version.

In summary, the paper establishes that for k = 5 the game always ends after exactly nine moves, and the second player possesses a deterministic winning strategy. The methodology—using convex layer types, beam regions, and systematic configuration transitions—offers a novel combinatorial‑geometric framework that could be extended to larger values of k or to other geometric avoidance games.