Computer solution to the 17-point Erdős-Szekeres problem

Szekeres, G.; Peters, Lindsay

doi:10.1017/s144618110000300x

Cited by 70 publications

(45 citation statements)

References 6 publications

(10 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Szekeres and Peters [28] proved, by an exhaustive computer search, that every set of 17 points in the plane, no three of which are collinear, contains 6 points in convex position. Note that it is easy to test whether a set of vertices is in convex position based on the directed crossing angle pattern α : E 2 → (0, π) ∪ { }.…”

Section: Complete Graphs Are Globally Crossing Angle Rigidmentioning

confidence: 99%

“…Proof: Denote by V the vertex set of G. Since all vertex pairs are adjacent, no three vertices are collinear in a straight-line drawing of G. Since |V | ≥ 17+6, we can successively choose two sets, P ⊂ V and Q ⊆ (V \ P ), each consisting of 6 points in convex position using [28]. Let conv(P ) = (p 1 , .…”

Section: Complete Graphs Are Globally Crossing Angle Rigidmentioning

confidence: 99%

See 1 more Smart Citation

Crossing Angles of Geometric Graphs

Arikushi¹,

Tóth²

2014

JGAA

View full text Add to dashboard Cite

We study the crossing angles of geometric graphs in the plane. We introduce the crossing angle number of a graph G, denoted can(G), which is the minimum number of angles between crossing edges in a straight-line drawing of G. We show that an n-vertex graph G with can(G) = O(1) has O(n) edges, but there are graphs G with bounded degree and arbitrarily large can(G). We also initiate the study of global crossing angle rigidity for geometric graphs. We construct bounded degree graphs G = (V, E) such that for any two straight-line drawings of G with the same crossing angle pattern, there is a subset V ⊂ V of |V | ≥ |V |/2 vertices that are embedded into similar point sets in the two drawings.

show abstract

Section: Complete Graphs Are Globally Crossing Angle Rigidmentioning

confidence: 99%

Section: Complete Graphs Are Globally Crossing Angle Rigidmentioning

confidence: 99%

Crossing Angles of Geometric Graphs

Arikushi¹,

Tóth²

2014

JGAA

View full text Add to dashboard Cite

show abstract

“…Recently, Koshelev [19] proved that H(6, 1) ≤ ES(7). Koshelev [18,19] also showed that Nyklová's proof is incorrect, and using a computer search based on the Szekeres-McKay-Peters algorithm [26] proved that H(6, k) = 17, for all 2 ≤ k ≤ 6 and H(6, 1) = 18. 1 Colored variants of the Erdős-Szekeres problem were considered by Devillers et al [8].…”

Section: Introductionmentioning

confidence: 99%

Almost empty monochromatic triangles in planar point sets

Basu

Bhattacharya

et al. 2016

Discrete Applied Mathematics

View full text Add to dashboard Cite

Abstract. For positive integers c, s ≥ 1, let M3(c, s) be the least integer such that any set of at least M3(c, s) points in the plane, no three on a line and colored with c colors, contains a monochromatic triangle with at most s interior points. The case s = 0, which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that M3(1, 0) = 3, M3(2, 0) = 9 and M3(c, 0) = ∞, for c ≥ 3. In this paper we extend these results when c ≥ 2 and s ≥ 1. We prove that the least integer λ3(c) such that M3(c, λ3(c)) < ∞ satisfies:where c ≥ 2. Moreover, the exact values of M3(c, s) are determined for small values of c and s. We also conjecture that λ3(4) = 1, and verify it for sufficiently large Horton sets.

show abstract

“…Nonetheless, as early as 1976, Appel and Haken proved the four color theorem [2,3] by using a computer program to check whether each of the thousands of possible candidates for the smallest-sized counter example to this theorem were actually four-colorable or not (simplified in [31]). Since then, important problems in various fields have been solved (fully or partially) with the assistance of computers: the discovery of Mersenne primes [39], the 17-point case of the happy ending problem [38], the NP-hardness of minimum-weight triangulation [26], a special case of Erdős' discrepancy conjecture [19], the ternary Goldbach conjecture [14], and Kepler's conjecture [13,24], among others. However, to the best of our knowledge, the scale of our computation is much greater than all of those and others which have been published.…”

Section: Introductionmentioning

confidence: 99%

Binary Pattern Tile Set Synthesis Is NP-hard

Kari

Kopecki

Meunier

et al. 2015

Automata, Languages, and Programming

View full text Add to dashboard Cite

In the field of algorithmic self-assembly, a long-standing unproven conjecture has been that of the NP-hardness of binary pattern tile set synthesis (2-Pats). The k-Pats problem is that of designing a tile assembly system with the smallest number of tile types which will self-assemble an input pattern of k colors. Of both theoretical and practical significance, k-Pats has been studied in a series of papers which have shown k-Pats to be NP-hard for k = 60, k = 29, and then k = 11. In this paper, we close the fundamental conjecture that 2-Pats is NP-hard, concluding this line of study.While most of our proof relies on standard mathematical proof techniques, one crucial lemma makes use of a computer-assisted proof, which is a relatively novel but increasingly utilized paradigm for deriving proofs for complex mathematical problems. This tool is especially powerful for attacking combinatorial problems, as exemplified by the proof of the four color theorem by Appel and Haken (simplified later by Robertson, Sanders, Seymour, and Thomas) or the recent important advance on the Erdős discrepancy problem by Konev and Lisitsa using computer programs. We utilize a massively parallel algorithm and thus turn an otherwise intractable portion of our proof into a program which requires approximately a year of computation time, bringing the use of computer-assisted proofs to a new scale. We fully detail the algorithm employed by our code, and make the code freely available online.

show abstract

Computer solution to the 17-point Erdős-Szekeres problem

Cited by 70 publications

References 6 publications

Crossing Angles of Geometric Graphs

Crossing Angles of Geometric Graphs

Almost empty monochromatic triangles in planar point sets

Binary Pattern Tile Set Synthesis Is NP-hard

Contact Info

Product

Resources

About