Ramsey Theory in the Work of Paul Erdős

Graham, Ron; Nešetřil, Jaroslav

doi:10.1007/978-3-642-60406-5_16

Cited by 3 publications

(3 citation statements)

References 82 publications

(88 reference statements)

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“…The paper [38] discusses the importance of [32] in the development of Ramsey theory. In the following theorem we give three different methods for getting upper bounds on N (n) based on Ramsey's theorem.…”

Section: Problem 14 ([27]) For Any Positive Integer N ≥ 3 Determinmentioning

confidence: 99%

The Erdos-Szekeres problem on points in convex position – a survey

Morris

Soltan

2000

Bull. Amer. Math. Soc.

102

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Abstract. In 1935 Erdős and Szekeres proved that for any integer n ≥ 3 there exists a smallest positive integer N (n) such that any set of at least N (n) points in general position in the plane contains n points that are the vertices of a convex n-gon. They also posed the problem to determine the value of N (n) and conjectured that N (n) = 2 n−2 + 1 for all n ≥ 3.Despite the efforts of many mathematicians, the Erdős-Szekeres problem is still far from being solved. This paper surveys the known results and questions related to the Erdős-Szekeres problem in the plane and higher dimensions, as well as its generalizations for the cases of families of convex bodies and the abstract convexity setting.

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Section: Problem 14 ([27]) For Any Positive Integer N ≥ 3 Determinmentioning

confidence: 99%

The Erdos-Szekeres problem on points in convex position – a survey

Morris

Soltan

2000

Bull. Amer. Math. Soc.

102

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“…For example, the planar 4-points-in-convex-position predicate mentioned above is unavoidable, since every generic five-point set contains four points in convex position. This result is known as the "happy ending problem", and it is actually one of the original results that motivated the development of Ramsey theory [GN13].…”

Section: Introductionmentioning

confidence: 96%

Classifying unavoidable Tverberg partitions

Bukh,

Loh,

Nivasch

2016

Preprint

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Let T (d, r) def = (r − 1)(d + 1) + 1 be the parameter in Tverberg's theorem, and call a partition I of {1, 2, . . . , T (d, r)} into r parts a Tverberg type. We say that I occurs in an ordered point sequence P if P contains a subsequence P of T (d, r) points such that the partition of P that is order-isomorphic to I is a Tverberg partition. We say that I is unavoidable if it occurs in every sufficiently long point sequence.In this paper we study the problem of determining which Tverberg types are unavoidable. We conjecture a complete characterization of the unavoidable Tverberg types, and we prove some cases of our conjecture for d ≤ 4. Along the way, we study the avoidability of many other geometric predicates.Our techniques also yield a large family of T (d, r)-point sets for which the number of Tverberg partitions is exactly (r − 1)! d . This lends further support for Sierksma's conjecture on the number of Tverberg partitions.

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“…It was also the topic of Ron's invited lecture at ICM 82 (held in Warsaw 1983) [Gra84]. Ramsey theory was also dear to Paul Erd ős as witnessed by the 2-volume set Mathematics of Paul Erdős where it occupies a whole chapter ([GNB13]; see also [GN97]). In fact these volumes, which were assembled under the guidance of Paul Erd ős himself, contain many pages written by the editors reflecting a long experience of collaboration with Erd ős.…”

Section: Ron and Ramseymentioning

confidence: 99%