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 avoidance game that is a natural game‑theoretic analogue of the Erdős‑Szekeres theorem. In the game, players A and B alternately place points in the Euclidean plane. The objective is to avoid creating a convex k‑gon among all points that have been placed; the player who places the point that completes the first convex k‑gon loses. The authors focus on the case k = 5, both for ordinary convex pentagons and for empty convex pentagons (pentagons whose interior contains no other points).

After a brief survey of the classical Erdős‑Szekeres problem, its well‑studied variants (empty convex k‑gons, interior‑point constraints, chromatic versions) and recent work on combinatorial games on point sets, the authors formalize the game. They define the convex hull of the current point set and the recursive convex layers (the “onion” decomposition) that arise when the hull is peeled away repeatedly. These layers are the central tool for analyzing the game because a convex k‑gon appears precisely when k points lie on the same outermost layer or when a suitable configuration of points across several layers forces a convex hull with k vertices.

The main contributions are twofold. First, the authors construct an explicit winning strategy for the second player (B) in the convex‑5‑gon game. The strategy proceeds by maintaining a “safe” configuration of convex layers after each of B’s moves. At each turn B examines the location chosen by A, determines how it would modify the current onion structure, and then places a point in a position that either (i) adds a new outer layer without creating a 5‑vertex hull, or (ii) fills an interior layer in such a way that the outer hull still has at most four vertices. By systematically enumerating all possible configurations after up to eight moves, the authors show that B can always respond so that the point set never contains a convex pentagon after his own move.

Second, the authors prove that the game must terminate on the ninth move, regardless of the choices made by the first player. They achieve this by a case‑analysis of all admissible layer configurations after eight points have been placed. There are twelve essentially distinct configurations (varying numbers of layers, sizes of each layer, and relative positions). For each configuration they identify the region of the plane where a ninth point could be placed without immediately forming a convex pentagon. In every case that region is empty or, if a point is placed there, it inevitably creates a convex 5‑gon either on the outer hull or by combining points from two layers. Consequently, after eight points the game is forced into a “dead end” where the next move inevitably loses, and because B moves second, B is guaranteed to be the player who does not place the fatal ninth point.

The same line of reasoning applies to the empty‑convex‑5‑gon variant. The crucial difference is that B must also prevent any interior point from lying inside a potential pentagon. The authors show that by always placing points on the current outer hull and never allowing an interior layer to contain more than one point, B can keep the configuration “empty” up to eight moves. The ninth move again forces a pentagon, and because the interior is empty by construction, the resulting pentagon is empty as well, satisfying the loss condition for the player who placed the ninth point.

The paper concludes with several directions for future work. Extending the analysis to larger k (k ≥ 6) is non‑trivial; the number of possible layer configurations grows rapidly, and it is unclear whether the second player retains a universal winning strategy. Introducing additional constraints—such as restricting points to integer lattice positions, imposing a coloring rule, or limiting the allowable region of placement—could dramatically change the game’s dynamics and may lead to new combinatorial bounds related to the classic Erdős‑Szekeres numbers. Finally, the authors suggest that the convex‑layer methodology used here could be adapted to other online geometric games, for example, avoidance games involving other convex shapes or higher‑dimensional analogues.

In summary, the paper provides a rigorous combinatorial‑geometric analysis of a novel two‑player game derived from the Erdős‑Szekeres theorem, establishes a definitive winning strategy for the second player in the convex‑5‑gon and empty‑convex‑5‑gon cases, and proves that the game inevitably ends on the ninth move. This work bridges classical extremal geometry with game theory and opens a fertile avenue for further exploration of geometric avoidance games.