Holes or Empty Pseudo-Triangles in Planar Point Sets

Let $E(k, \ell)$ denote the smallest integer such that any set of at least $E(k, \ell)$ points in the plane, no three on a line, contains either an empty convex polygon with $k$ vertices or an empty pseudo-triangle with $\ell$ vertices. The existence of $E(k, \ell)$ for positive integers $k, \ell\geq 3$, is the consequence of a result proved by Valtr [Discrete and Computational Geometry, Vol. 37, 565–576, 2007]. In this paper, following a series of new results about the existence of empty pseudo-triangles in point sets with triangular convex hulls, we determine the exact values of $E(k, 5)$ and $E(5, \ell)$, and prove bounds on $E(k, 6)$ and $E(6, \ell)$, for $k, \ell\geq 3$. By dropping the emptiness condition, we define another related quantity $F(k, \ell)$, which is the smallest integer such that any set of at least $F(k, \ell)$ points in the plane, no three on a line, contains a convex polygon with $k$ vertices or a pseudo-triangle with $\ell$ vertices. Extending a result of Bisztriczky and T'oth [Discrete Geometry, Marcel Dekker, 49–58, 2003], we obtain the exact values of $F(k, 5)$ and $F(k, 6)$, and obtain non-trivial bounds on $F(k, 7)$.

💡 Research Summary

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The paper investigates two extremal functions defined on planar point sets in general position (no three collinear). The first function, (E(k,\ell)), is the smallest integer such that any set of at least (E(k,\ell)) points contains either an empty convex polygon with (k) vertices (an “empty hole”) or an empty pseudo‑triangle with (\ell) vertices. The second function, (F(k,\ell)), drops the emptiness requirement and asks for the smallest integer guaranteeing a convex (k)-gon or a (not necessarily empty) pseudo‑triangle with (\ell) vertices.

Valtr (2007) proved that (E(k,\ell)) is finite for all (k,\ell\ge 3), but gave no concrete values. Bisztriczky and Tóth (2003) studied the non‑empty version and obtained bounds for small (\ell). The present work builds on these foundations and provides exact values for several families of parameters, together with non‑trivial upper and lower bounds for the remaining cases.

Main Contributions

1. Exact values for (E(k,5)) and (E(5,\ell)).
   The authors show that
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