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 estimateby 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.
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.
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
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:
not identically zero;
for every permutation and indices ,
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
-dimensional vectorsand 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:
for every permutation on any distinct indices ,
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
CaDiCaLmanaged to prove unsatisfiability in about 2 CPU days. Moreover, the unsatisfiability certificate created by
CaDiCaLis about 39 GB and the
DRAT-trimverification took about 1 CPU day.
: The size of the instance is about 433 MB and
--unsat) managed to prove unsatisfiability in about 19 CPU days.
: The size of the instance is about 955 MB and
CaDiCaLmanaged to prove unsatisfiability in about 7 CPU days.
: The size of the instance is about 4.2 GB and
CaDiCaLmanaged to prove unsatisfiability in about 3 CPU days. Moreover, the unsatisfiability certificate created by
CaDiCaLis about 117 GB and the
DRAT-trimverification 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].
Unfortunately, the unsatisfiability certificates for
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]).