A SAT attack on higher dimensional Erdős–Szekeres numbers

05/18/2021 ∙ by Manfred Scheucher, et al. ∙ Berlin Institute of Technology (Technische Universität Berlin) 0

A famous result by Erdős and Szekeres (1935) asserts that, for every k,d ∈ℕ, there is a smallest integer n = g^(d)(k), such that every set of at least n points in ℝ^d in general position contains a k-gon, i.e., a subset of k points which is in convex position. We present a SAT model for higher dimensional point sets which is based on chirotopes, and use modern SAT solvers to investigate Erdős–Szekeres numbers in dimensions d=3,4,5. We show g^(3)(7) ≤ 13, g^(4)(8) ≤ 13, and g^(5)(9) ≤ 13, which are the first improvements for decades. For the setting of k-holes (i.e., k-gons with no other points in the convex hull), where h^(d)(k) denotes the minimum number n such that every set of at least n points in ℝ^d in general position contains a k-hole, we show h^(3)(7) ≤ 14, h^(4)(8) ≤ 13, and h^(5)(9) ≤ 13. Moreover, all obtained bounds are sharp in the setting of chirotopes and we conjecture them to be sharp also in the original setting of point sets.

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1 Introduction

The classical Erdős–Szekeres Theorem [ErdosSzekeres1935] asserts that every sufficiently large point set in the plane in general position (i.e., no three points on a common line) contains a -gon (i.e., a subset of points in convex position).

Theorem 1 ([ErdosSzekeres1935], The Erdős–Szekeres Theorem).

For every integer , there is a smallest integer such that every set of at least points in general position in the plane contains a -gon.

Erdős and Szekeres showed that [ErdosSzekeres1935] and constructed point sets of size without -gons [ErdosSzekeres1960], which they conjectured to be extremal. There were several improvements of the upper bound in the past decades, each of magnitude , and in 2016, Suk showed [Suk2017]. Shortly after, Holmsen et al. [HolmsenMPT2020] slightly improved the error term in the exponent and presented a generalizion to chirotopes (see Section 2 for a definition). The lower bound is known to be sharp for . The value was determined by Klein in 1935, was determined by Makai (cf. Kalbfleisch et al. [KalbfleischKalbfleischStanton1970]), and was shown by Szekeres and Peters [SzekeresPeters2006] using heavy computer assistance. While their computer program uses thousands of CPU hours, we have developed a SAT framework [Scheucher2020_CGTA] which allows to verify this result within only 2 CPU hours, and an independent verification of their result using SAT solvers was done by Marić [Maric2019].

1.1 Planar -Holes

In the 1970’s, Erdős [Erdos1978] asked whether every sufficiently large point set contains a -hole, that is, a -gon with the additional property that no other point lies in its convex hull. In the same vein as , we denote by the smallest integer such that every set of at least points in general position in the plane contains a -hole. This variant differs significantly from the original setting as there exist arbitrarily large point sets without 7-holes [Horton1983] (cf. [Valtr1992a]). While Harborth [Harborth1978] showed , the existence of 6-holes remained open until 2006, when Gerken [Gerken2008] and Nicolás [Nicolas2007] independently showed that sufficiently large point sets contain 6-holes (cf. [Valtr2009]). Today the best bounds concerning 6-holes are . The lower bound is witnessed by a set of 29 points without 6-holes that was found via a simulated annealing approach by Overmars [Overmars2002], and the proof of upper bound by Koshelev [Koshelev2009] covers 50 pages (written in Russian). To be more specific, similar to Gerken, who used 9-gons to find 6-holes [Gerken2008]

, Koshelev used 8-gons with many interior points to find 6-holes. Using the estimate

by Tóth and Valtr [TothValtr2004], he then concluded .

1.2 Higher Dimensions

The notions general position (no

points in a common hyperplane),

-gon (a set of points in convex position), and -hole (a -gon with no other points in the convex hull) naturally generalize to higher dimensions, and so does the Erdős–Szekeres Theorem [ErdosSzekeres1935, DanzerGK1963] (cf. [MorrisSoltan2000]). We denote by and the minimum number of points in in general position that guarantee the existence of -gon and -hole, respectively. In contrast to the planar case, the asymptotic behavior of the higher dimensional Erdős–Szekeres numbers remains unknown for dimension . While a dimension-reduction argument by Károlyi [Karolyi2001] combined with Suk’s bound [Suk2017] shows

for , the currently best asymptotic lower bound is with is witnessed by a construction by Károlyi and Valtr [KarolyiValtr2003]. For dimension 3, Füredi conjectured (unpublished, cf. [Matousek2002_book, Chapter 3.1]).

1.3 Higher dimensional holes

Since Valtr [Valtr1992b] gave a construction for any dimension without -holes, generalizing the idea of Horton [Horton1983], the central open problem about higher dimensional holes is to determine the largest value such that every sufficiently large set in -space contains a -hole. Note that with this notation we have because [Gerken2008, Nicolas2007] and [Horton1983]. Very recently Bukh et al. [BukhChaoHolzman2020] presented a construction without -holes, which further improves Valtr’s bound and shows . On the other hand, the dimension-reduction argument by Károlyi [Karolyi2001] also applies to -holes, and therefore

This inequality together with implies that and hence . However, already in dimension 3 the gap between the upper and the lower bound of remains huge: while there are arbitrarily large sets without 23-holes [Valtr1992b], already the existence of 8-holes remains unknown ().

1.4 Precise Values

As discussed before, for the planar -gons , , , and are known. For planar -holes, , , and for .

While the values for and are easy to determine (cf. [BisztriczkySoltan1994]), Bisztriczky et al. [BisztriczkySoltan1994, BisztriczkyHarborth1995, MorrisSoltan2000] showed for . This, in particular, determines the values for and shows . For and , Bisztriczky and Soltan [BisztriczkySoltan1994] moreover determined the values . Tables 1 and 2 summarize the currently best bounds for -gons and -holes in small dimensions.

*
*
*
Table 1: Known values and bounds for . Entries marked with a star (*) are new. Entries left blank can be upper-bounded by the estimate [TothValtr2004] and the dimension-reduction argument from [Karolyi2001].
* ? ? ? ?
* ? ? ?
* ? ?
?
Table 2: Known values and bounds for . Entries marked with a star (*) are new.

1.5 Our Results

In this article we show the following upper bounds on higher dimensional Erdős–Szekeres numbers and the -holes variant in dimensions 3, 4 and 5, which we moreover conjecture to be sharp.

Theorem 2.

It holds , , , and .

For the proof of Theorem 2, we generalize our SAT framework from [Scheucher2020_CGTA] to higher dimensional point sets. Our framework for dimensions is based on chirotopes of rank , and we use the SAT solver CaDiCaL [Biere2019] to prove unsatisfiability. Moreover, CaDiCaL can generate unsatisfiability proofs which then can be verified by a proof checking tool such as DRAT-trim [WetzlerHeuleHunt2014].

2 Preliminaries

Let be a set of labeled points in in general position with coordinates . We assign to each -tuple a sign to indicate whether the corresponding points are positively or negatively oriented. Formally, we define a mapping with

It is well known that this mapping is a chirotope of rank (cf. [BjoenerLVWSZ1993, Definition 3.5.3]).

Definition 1 (Chirotope).

A mapping is a chirotope of rank  if the following three properties are fulfilled:

  1. not identically zero;

  2. for every permutation and indices ,

  3. for indices ,

Note that the mapping fulfills the first axiom of Definition 1 because is induced the point set , which is in general position. The second axiom is fulfilled because, by the properties of the determinant, we have

for any

-dimensional vectors

and any permutation of the indices . Since we can consider the homogeneous coordinates of our -dimensional point set as -dimensional vectors, the above relation also has to be respected by . To see that also fulfills the third axiom, recall that the well-known Graßmann–Plücker relations (see e.g. [BjoenerLVWSZ1993, Chapter 3.5]) assert that any -dimensional vectors fulfill111 The Graßmann–Plücker relations can be derived using Linear Algebra basics as outlined: Consider the vectors as an matrix and apply row additions to obtain the echelon form. If the first columns form a singular matrix, then and both sides of the equation vanish by a simple column multiplication argument. Otherwise, we can assume that the first columns form an identify matrix. Since the determinant is invariant to row additions, none of the terms in the Graßmann–Plücker relations is effected during the transformation, and the statement then follows from Laplace expansion.

In particular, if all summands on the right-hand side are non-negative then also the left-hand side must be non-negative.

In Section 3 we will model chirotopes in a SAT model. While the axioms from Definition 1 require constraints, we can significantly reduce this number to by using an axiom system based on the 3-term Graßmann–Plücker relations.

Theorem 3 (3-Term Graßmann–Plücker relations,  [BjoenerLVWSZ1993, Theorem 3.6.2]).

A mapping is a non-degenerate chirotope of rank  if the following two properties are fulfilled:

  1. for every permutation on any distinct indices ,

  2. for any ,

2.1 Gons and Holes

Carathéodory’s theorem asserts that a -dimensional point set is in convex position if and only if all -element subsets are in convex position. Now that a point lies in the convex hull of if and only if holds for every , we can fully axiomize -gons and -holes solely using the information of the chirotope, that is, the relative position of the points. (The explicit coordinates do not play a role.)

3 The SAT Framework

For the proof of Theorem 2, we proceed as following: To show (or , resp.), assume towards a contradiction that there exists a set of points in in general position, which does not contain any -gon (or -hole, resp.). The point set induces a chirotope of rank , which can be encoded using Boolean variables. The chirotope fulfills the conditions from Theorem 3, which we can encode as clauses.

Next, we introduce auxiliary variables for all to indicate whether the point lies in the convex hull of . As discussed in Section 1.2, the values of these auxiliary variables are fully determined by the chirotope (variables). Using these auxiliary variables we can formulate clauses, each involving literals, to assert that there are no -gons in : Among every subset of size there is at least one point which is contained in the convex hull of points of . (To assert that there are no -holes in , we can proceed in a similar manner: Among every subset of size there is at least one point which is contained in the convex hull of points of .)

Altogether, we can now create a Boolean satisfiability instance that is satisfiable if and only if there exists a rank chirotope on elements without -gons (or -holes, resp.). If the instance is provable unsatisfiable, no such chirotope (and hence no point set ) exists, and we have (or , resp.).

3.1 Running Times and Resources

All our computations were performed on single CPUs. However, since some computations (especially for verifying the unsatisfiability certificates) required more resources than available on standard computers/laptops, we made use of the computing cluster from the Institute of Mathematics at TU Berlin.

  • : The size of the instance is about 245 MB and CaDiCaL managed to prove unsatisfiability in about 2 CPU days. Moreover, the unsatisfiability certificate created by CaDiCaL is about 39 GB and the DRAT-trim verification took about 1 CPU day.

  • : The size of the instance is about 433 MB and CaDiCaL (with parameter --unsat) managed to prove unsatisfiability in about 19 CPU days.

  • : The size of the instance is about 955 MB and CaDiCaL managed to prove unsatisfiability in about 7 CPU days.

  • : The size of the instance is about 4.2 GB and CaDiCaL managed to prove unsatisfiability in about 3 CPU days. Moreover, the unsatisfiability certificate created by CaDiCaL is about 117 GB and the DRAT-trim verification took about 3 CPU days.

The python program for creating the instances and further technical information is available on our supplemental website [website_ES_highdim].

4 Discussion

Unfortunately, the unsatisfiability certificates for and , respectively, created by CaDiCaL grew too big to be verifiable with our available resources. However, it might be possible to further optimize the SAT model to make the solver terminate faster (cf. [Scheucher2020_CGTA]) so that one obtains smaller certificates.

In the course of our investigations we found chirotopes that witness that all bounds from Theorem 2 are sharp in the more general setting of chirotopes. However, since we have not yet succeeded in finding realizations of those chirotopes, we can only conjecture that all bounds from Theorem 2 are also sharp in the original setting, but we are looking forward to implementing further computer tools so that we can address all those realizability issues. It is worth noting that finding realizable witnesses is a notoriously hard and challenging task because (i) only of the rank chirotopes are realizable by point sets and (ii) the problem of deciding realizability is ETR-complete in general (cf. Chapters 7.4 and 8.7 in [BjoenerLVWSZ1993]).

Concerning the existence of 8-holes in 3-space: while we managed to find a rank 4 chirotope on 18 elements without 8-holes within only a few CPU hours, the solver did not terminate for months on the instance . We see this as a strong evidence that sufficiently large sets in 3-space (possibly already 19 points suffice) contain 8-holes.

Last but not least our SAT framework can also be used to tackle other problems on higher dimensional point sets. By slightly adapting our SAT framework, we managed to answer a Tverberg-type question by Fulek et al. (cf. Section 3.2 in [FulekGKVW2019]).

References