Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique” conjecture of K'ara, P'or, and Wood [\emph{Discrete Comput. Geom.} 34(3):497–506, 2005].

💡 Research Summary

The paper establishes a far‑reaching generalisation of the classic empty‑pentagon theorem. For any integer ℓ ≥ 2 there exists a threshold N(ℓ) such that every planar point set S with |S| ≥ N(ℓ) must contain either ℓ collinear points or an empty pentagon (a five‑vertex convex polygon whose interior is free of points of S). This dichotomy extends the well‑known result that sufficiently large point sets contain an empty quadrilateral (the case ℓ = 3) and provides the first non‑trivial bound for arbitrary ℓ.

The proof proceeds in two complementary regimes, distinguished by the local density of the point set. The authors first introduce a grid‑based decomposition of the plane into equal‑sized squares (cells) and define a density parameter τ that bounds the number of points per cell. If any cell contains more than τ points, a combinatorial‑geometric argument based on the pigeon‑hole principle and a careful analysis of minimal distances yields a construction of ℓ collinear points. This part of the argument relies on a “density‑implies‑collinearity” lemma: a cell that is sufficiently crowded must contain an ℓ‑point line segment, which is proved by iteratively extending a line through the densest pair of points until ℓ points are aligned.

When every cell respects the τ‑bound, the point set is globally sparse. In this regime the authors turn to a graph‑theoretic representation: they consider the complete geometric graph on S, label each edge as “empty” if the open segment contains no other point of S, and study the structure of empty cycles. By extending the classical empty‑quadrilateral theorem, they prove that if the graph contains no empty 4‑cycle (i.e., no empty quadrilateral), then it must contain an empty 5‑cycle, which corresponds precisely to an empty pentagon. The key technical steps are two auxiliary lemmas: (1) every empty 5‑cycle contains at least one empty triangle, and (2) any empty triangle can be “expanded’’ into an empty pentagon by a sequence of point‑exchange operations that preserve emptiness. The authors give a constructive proof of these lemmas, showing how to locate the necessary vertices using order‑type arguments and the Erdős‑Szekeres monotone subsequence theorem.

A quantitative version of the main theorem is derived by explicitly bounding τ and the cell size in terms of ℓ. The resulting threshold N(ℓ) is polynomial in ℓ, and the authors provide concrete constants for ℓ = 4, 5, 6 obtained through computational experiments on random point sets and on Horton’s classic construction that avoids empty quadrilaterals. Their implementation confirms that the theoretical bounds are not overly pessimistic.

The paper’s most immediate application is to the “big line or big clique’’ conjecture of Kára, Pór, and Wood (Discrete Comput. Geom. 34(3):497–506, 2005). That conjecture asserts that for any integers ℓ, k there exists a number f(ℓ,k) such that every point set of size at least f(ℓ,k) contains either ℓ collinear points (a “big line’’) or k points that form a complete geometric graph with all edges empty (a “big clique’’). By setting k = 5 and using the empty‑pentagon theorem proved here, the authors resolve the next open case of the conjecture: for every ℓ ≥ 2 there is a concrete f(ℓ,5) = N(ℓ) guaranteeing a big line of size ℓ or a 5‑clique consisting of the vertices of an empty pentagon. The proof shows that the vertices of an empty pentagon are pairwise connected by empty edges, thus forming the required clique.

Beyond this specific application, the work introduces a powerful methodological framework that blends density arguments, grid decompositions, and empty‑edge cycle analysis. The authors discuss several promising extensions: (i) simultaneous growth of both parameters ℓ and k, (ii) generalisation to higher dimensions where empty simplices replace empty polygons, and (iii) algorithmic aspects, including an O(n log n) procedure to detect either an ℓ‑collinear subset or an empty pentagon in a given point set. The paper concludes by highlighting open problems such as determining the exact asymptotic behaviour of N(ℓ) and exploring whether similar dichotomies hold for empty hexagons or higher‑order empty convex polygons.

In summary, the authors prove that any sufficiently large planar point set must exhibit one of two highly structured configurations—ℓ collinear points or an empty pentagon—thereby advancing the theory of empty convex polygons, providing a decisive step toward the big line/big clique conjecture, and laying groundwork for future investigations in discrete geometry and combinatorial optimization.