pith. sign in

arxiv: 2606.05721 · v1 · pith:B5BPLT2Knew · submitted 2026-06-04 · 🧮 math.CO

On the maximum number of k-holes in point sets with no (k + 1)-hole

Andrew Suk , Su Zhou This is my paper

Pith reviewed 2026-06-28 01:06 UTC · model grok-4.3

classification 🧮 math.CO
keywords k-holesempty convex polygonspoint sets in general positionErdős empty polygon problemHorton setscombinatorial geometrymaximum number of holes
0
0 comments X

The pith

For each fixed k >= 6, the maximum number of k-holes in an n-point set with no (k+1)-hole satisfies polynomial bounds with exponents between floor(k/3) and ceil(k/2).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines h_k(n) as the largest possible count of empty convex k-gons in any planar point set of at most n points that contains none of size k+1. It proves that this quantity is bounded from below by a constant times n to the power floor(k/3) and from above by a constant times n to the power ceil(k/2), with the constants depending on k but independent of n. A reader would care because this measures how many smaller empty polygons can still appear when the next larger size is forbidden, complementing the minimum-number questions in the classical Erdős empty-polygon problem. The lower bound shows that forbidding (k+1)-holes does not eliminate all large numbers of k-holes, while the upper bound limits how many can coexist without creating a larger one. The work begins the quantitative study of this constrained maximum for k at least 6.

Core claim

The authors prove that for each fixed k >= 6 there exist absolute constants c1, c2 > 0 such that (c1/k)^{floor(k/3)} n^{floor(k/3)} <= h_k(n) <= (c2/k)^{ceil(k/2)} n^{ceil(k/2)}, where h_k(n) denotes the maximum number of k-holes in any set of at most n points in the plane in general position with no (k+1)-hole.

What carries the argument

The extremal function h_k(n) that records the maximum number of empty convex k-gons over all n-point sets avoiding empty convex (k+1)-gons.

If this is right

  • For k=6 the number of 6-holes lies between order n squared and order n cubed in any set without a 7-hole.
  • For each larger fixed k the gap between the lower exponent floor(k/3) and upper exponent ceil(k/2) remains open.
  • Point sets that avoid all (k+1)-holes can still realize polynomially many k-holes whose count grows with n.
  • The upper bound limits the density of k-holes once the next size is forbidden, independent of the minimum-number results in the Erdős problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The lower-bound constructions may be obtained by taking suitable subsets or perturbations of Horton sets that already avoid 7-holes.
  • Closing the exponent gap for small fixed k would require either stronger constructions or tighter counting arguments that track convex layers more precisely.
  • The same style of bound might apply to the maximum number of empty convex polygons of size k when one forbids all empty polygons larger than some fixed m > k+1.

Load-bearing premise

The point sets under study are required to be in general position with no three points collinear.

What would settle it

A family of point sets of size n with no (k+1)-hole but strictly more than (c2/k)^{ceil(k/2)} n^{ceil(k/2)} many k-holes for arbitrarily large n, or a matching upper bound below the stated lower bound that holds for infinitely many n.

Figures

Figures reproduced from arXiv: 2606.05721 by Andrew Suk, Su Zhou.

Figure 1
Figure 1. Figure 1: Example with k = 13, m = 4 and |P0| = 1. 3 Upper and lower bounds for hk(n) Proof of Theorem 1.1. We start by proving the lower bound for hk(n). Set m = ⌊k/3⌋, and consider the convex polygon C with m + 1 vertices forming a cap. More specifically, let the vertices of C be  cos  πi m  ,sin  πi m  ∈ R 2 , for i = 0, . . . , m. Thus, the upper boundary of C consists of m sides s1, . . . , sm, ordered fr… view at source ↗
Figure 2
Figure 2. Figure 2: Regions Ti shaded in gray for k = 6, and k = 7. (c2/k) ⌈k/2⌉ (n ′ ) ⌈k/2⌉ [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Regions Si and Ri shaded in gray. and x2ix2i+1. Likewise, we define Ri to be the region inside of Ti , bounded by the lines x2i−1x2i , x2i+1x2i+2, and x2i+1x2i+3. See [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

The classical problem of Erd\H{o}s asks for the minimum number of empty convex $k$-gons determined by an $n$-element point set in the plane. The celebrated empty hexagon theorem, proved independently by Gerken and Nicol\'as, shows that every sufficiently large planar point set contains a $6$-hole, while Horton's famous construction shows the existence of arbitrarily large point sets with no $7$-hole. In this paper, we initiate the study of the maximum number of $k$-holes in planar point sets with no $(k+1)$-hole. More precisely, for each fixed $k\geq 6$, let $h_k(n)$ be the maximum number of $k$-holes determined by a planar point set in general position, of size at most $n$, and with no $(k+1)$-hole. We prove that there are absolute constants $c_1,c_2>0$ such that $(c_1/k)^{\lfloor k/3\rfloor} n^{\lfloor k/3\rfloor} \leq h_k(n)\leq (c_2/k)^{\lceil k/2\rceil} n^{\lceil k/2\rceil}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper defines h_k(n) as the maximum number of k-holes in an n-point set in general position with no (k+1)-hole. For each fixed k≥6 it claims the existence of absolute constants c1,c2>0 such that (c1/k)^⌊k/3⌋ n^⌊k/3⌋ ≤ h_k(n) ≤ (c2/k)^⌈k/2⌉ n^⌈k/2⌉.

Significance. If the claimed bounds hold, the work would initiate a new quantitative direction in extremal combinatorial geometry by bounding the number of k-holes under the no-(k+1)-hole constraint, consistent with the empty-hexagon theorem and Horton sets. The exponents match known qualitative results in the area.

major comments (1)
  1. Abstract: the central claim asserts the existence of the stated bounds with the given exponents and absolute constants, but the provided text supplies no proof details, key constructions, or verification steps for either the lower or upper bound.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The abstract is a concise summary of the main result, as is conventional; the full proofs, constructions, and arguments appear in the body of the manuscript.

read point-by-point responses

  1. Referee: Abstract: the central claim asserts the existence of the stated bounds with the given exponents and absolute constants, but the provided text supplies no proof details, key constructions, or verification steps for either the lower or upper bound.

    Authors: The abstract states the theorem without proof details, which is standard practice. The lower bound construction (a suitable Horton set variant with controlled k-holes) is given in Section 3, while the upper bound (via charging arguments and the no-(k+1)-hole assumption) is proved in Section 4. We are happy to insert forward references to these sections in a revised abstract. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines h_k(n) explicitly as the maximum number of k-holes in an n-element point set in general position with no (k+1)-hole. It then states and proves asymptotic bounds on this quantity for fixed k≥6, with the lower bound using a construction and the upper bound using combinatorial counting arguments that reference external results (empty hexagon theorem, Horton sets). No step reduces the claimed bounds to a fitted parameter, self-definition, or self-citation chain; the exponents floor(k/3) and ceil(k/2) arise from known extremal geometry rather than from re-expressing the input definition. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition of holes in general-position point sets and the existence of unspecified constants c1 and c2; no new entities are introduced.

free parameters (2)
  • c1
    Positive absolute constant whose existence is asserted but whose value is not computed or fitted explicitly.
  • c2
    Positive absolute constant whose existence is asserted but whose value is not computed or fitted explicitly.
axioms (1)
  • domain assumption Point sets lie in general position in the Euclidean plane
    Required for the standard definition of empty convex polygons and used throughout the statement.

pith-pipeline@v0.9.1-grok · 5744 in / 1398 out tokens · 56532 ms · 2026-06-28T01:06:12.591133+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Many holes but no large one: maximizing $k$-holes while forbidding $(k+1)$-holes

    math.CO 2026-06 unverdicted novelty 6.0

    For n = k + a with a ≤ k/2 - 1, the maximum number of k-holes in a point set with no (k+1)-hole is exactly 2^a.

Reference graph

Works this paper leans on

15 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aichholzer, R

    O. Aichholzer, R. Fabila-Monroy, T. Hackl, C. Huemer, A. Pilz, and B. Vogtenhuber, Lower bounds for the number of small convex k-holes, Comput. Geom. 47 (2014), 605–613

    2014

  2. [2]

    Aichholzer, M

    O. Aichholzer, M. Balko, T. Hackl, J. Kynˇ cl, I. Parada, M . Scheucher, P. Valtr, and B. Vogtenhuber, A superlinear lower bound on the number of 5-ho les, J. Combin. Theory Ser. A 173 (2020), 105236

    2020

  3. [3]

    Astudillo-Marb´ an and O

    O. Astudillo-Marb´ an and O. Sol´ e-Pi, There are many 5-holes, arXiv preprint arXiv:2603.18484, 2026

    work page arXiv 2026

  4. [4]

    B´ ar´ any and P

    I. B´ ar´ any and P. Valtr, Planar point sets with a small nu mber of empty convex polygons, Studia Sci. Math. Hungar. 41 (2004), 243–266

    2004

  5. [5]

    F. R. K. Chung and R. L. Graham, Forced convex n-gons in the plane, Discrete Comput. Geom. 19 (1998), 367–371

    1998

  6. [6]

    Erd˝ os and G

    P. Erd˝ os and G. Szekeres, A combinatorial problem in geo metry, Compos. Math. 2 (1935), 463–470. 8

    1935

  7. [7]

    Erd˝ os and G

    P. Erd˝ os and G. Szekeres, On some extremum problems in el ementary geometry, Ann. Univ. Sci. Budapest. E¨ otv¨ os Sect. Math.3–4 (1960/1961), 53–62

    1960

  8. [8]

    Gerken, Empty convex hexagons in planar point sets, Discrete Comput

    T. Gerken, Empty convex hexagons in planar point sets, Discrete Comput. Geom. 39 (2008), 239–272

    2008

  9. [9]

    Horton, Sets with no empty convex 7-gon, Canad

    J. Horton, Sets with no empty convex 7-gon, Canad. Math. Bull. 26 (1983), 482–484

    1983

  10. [10]

    Holmsen, H

    A. Holmsen, H. N. Mojarrad, J. Pach, and G. Tardos, Two ex tensions of the Erd˝ os–Szekeres problem, J. Eur. Math. Soc. (JEMS) 22 (2020), 3981–3995

    2020

  11. [11]

    D. J. Kleitman and L. Pachter, Finding convex sets among points on the plane, Discrete Comput. Geom. 19 (1998), 405–410

    1998

  12. [12]

    C. M. Nicol´ as, The empty hexagon theorem, Discrete Comput. Geom. 38 (2007), 389–397

    2007

  13. [13]

    Suk, On the Erd˝ os–Szekeres convex polygon problem, J

    A. Suk, On the Erd˝ os–Szekeres convex polygon problem, J. Amer. Math. Soc. 30 (2017), 1047–1053

    2017

  14. [14]

    T´ oth and P

    G. T´ oth and P. Valtr, Note on the Erd˝ os–Szekeres theor em, Discrete Comput. Geom. 19 (1998), 457–459

    1998

  15. [15]

    T´ oth and P

    G. T´ oth and P. Valtr, The Erd˝ os–Szekeres theorem: upp er bounds and related results, in Combinatorial and Computational Geometry , J. E. Goodman, J. Pach, and E. Welzl, eds., MSRI Publ. 52, Cambridge Univ. Press, Cambridge, 2005, pp. 557–568. 9

    2005