Forced projective 7-holes in sufficiently large point sets in the real projective plane

In the Euclidean plane the largest $k$ for which every sufficiently large point set in general position must contain an empty convex $k$ -gon (a $k$ -hole) is exactly 6: Horton constructed arbitrarily large sets with no 7-hole, while Gerken and Nicolás proved that 6-holes are unavoidable. Balko, Scheucher, and Valtr transplanted this question to the real projective plane $\mathbb{R}P^{2}$ using Steinitz's 1913 notion of convexity, where a projective $k$ -hole is, in a suitable affine chart, either an empty convex $k$ -gon or an empty region bounded by two concave chains. They proved that projective $k$ -holes are forced for $k\le 6$ and constructed arbitrarily large sets with no projective 8-hole, leaving a single undecided value: their Conjecture 1 asserts that every sufficiently large point set in $\mathbb{R}P^{2}$ contains a projective 7-hole. No resolution has been found as of August 2026; the problem is open, equivalently stated as deciding whether $h^{p}_{7}(n)>0$ for all sufficiently large $n$ , i.e. whether the projective forced-hole threshold $H^{p}$ equals 6 or 7.

Background

In 1935 Erdős and Szekeres proved that for every integer $k\ge 3$ there is a smallest number $ES(k)$ such that any planar point set in general position (no three points on a common line) with at least $ES(k)$ points contains $k$ points in convex position, that is, forming the vertex set of a convex $k$ -gon. A strengthening asks for such $k$ -gons with empty interior: a $k$ -hole in a finite point set $S$ is a set of $k$ points of $S$ in convex position whose convex hull contains no point of $S$ in its interior. Erdős asked in 1978 for the largest $k$ such that every sufficiently large planar point set in general position contains a $k$ -hole. Horton settled the upper side in 1983 by constructing arbitrarily large point sets with no 7-hole, and Gerken and Nicolás independently proved that every sufficiently large planar point set contains a 6-hole; in the Euclidean plane the answer is therefore exactly 6, a benchmark result of combinatorial geometry.

Balko, Scheucher, and Valtr transferred these questions to the real projective plane $\mathbb{R}P^{2}$ , the surface obtained from the Euclidean plane by adding one point at infinity for each direction, in which the complement of any projective line is a Euclidean (affine) plane. A finite set $P\subset\mathbb{R}P^{2}$ is in general position if no three of its points lie on a common projective line. Convexity in $\mathbb{R}P^{2}$ uses a definition going back to Steinitz (1913): a subset $X$ is semiconvex if any two of its points can be joined by a line segment contained in $X$ , and $X$ is convex if it is semiconvex and is contained in some affine plane $\rho\subset\mathbb{R}P^{2}$ (equivalently, it avoids some projective line). A projective convex hull of a set $Y$ is an inclusion-wise minimal convex subset of $\mathbb{R}P^{2}$ containing $Y$ ; unlike in the Euclidean plane, it need not be unique. A projective $k$ -gon determined by $P$ is a projective convex hull of $k$ points of $P$ that contains all of them on its boundary, and a projective $k$ -hole is such a $k$ -gon whose interior contains no point of $P$ . In an affine chart, a projective $k$ -hole is either an ordinary empty convex $k$ -gon or an empty double-chain $k$ -wedge, a region bounded by two concave chains meeting at their ends — a configuration with no analogue in the Euclidean theory. These notions were introduced following Harborth and Möller's 1994 study of projective $k$ -gons (the Esther Klein problem in the projective plane), and they make $\mathbb{R}P^{2}$ a new arena for Erdős–Szekeres-type extremal questions.

Balko, Scheucher, and Valtr proved that there are arbitrarily large point sets in $\mathbb{R}P^{2}$ with no projective 8-hole, using a construction based on Horton sets (a classical recursively defined family of planar point sets, generalized by Valtr, whose points lie in deep nested 'wedges' so that large empty convex polygons cannot form). Hence projective $k$ -holes are not forced for any $k\ge 8$ . On the other side, the theorem of Gerken and Nicolás implies that every sufficiently large point set in $\mathbb{R}P^{2}$ contains a projective $k$ -hole for every $k\le 6$ , since an affine $k$ -hole is in particular a projective $k$ -hole. The largest forced hole size in the real projective plane is therefore either 6 or 7, and the obvious affine attack on the remaining case fails: the very Horton sets that avoid affine 7-holes provably contain $\Theta(n^{2})$ projective 7-holes, so the affine counterexample technique does not extend.

Problem Statement

Determine whether there exists an integer $N$ such that every finite set of at least $N$ points in general position in the real projective plane $\mathbb{R}P^{2}$ contains a projective 7-hole. Equivalently, writing $h^{p}_{7}(n)$ for the minimum number of projective 7-holes over all $n$ -point sets in general position in $\mathbb{R}P^{2}$ , determine whether $h^{p}_{7}(n)>0$ for all sufficiently large $n$ . A negative answer must exhibit, for arbitrarily large $n$ , an $n$ -point set in general position in $\mathbb{R}P^{2}$ with no projective 7-hole. Either outcome closes the gap $6\le H^{p}\le 7$ left by Balko, Scheucher, and Valtr and determines $H^{p}$ , the largest $k$ such that every sufficiently large point set in $\mathbb{R}P^{2}$ contains a projective $k$ -hole. This is Conjecture 1 of Balko–Scheucher–Valtr, who conjecture an affirmative answer.

References

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