Monochromatic Schur triples in randomly perturbed dense sets of integers

Elad Aigner-Horev; Yury Person

doi:10.1137/18M1227007

Monochromatic Schur triples in randomly perturbed dense sets of integers

Elad Aigner-Horev
, Yury Person

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

Given a dense subset A of the first n positive integers, we provide a short proof showing that for p = ω(n^-2/3), the so-called randomly perturbed set A∩ [n]_p a.a.s. has the property that any 2-coloring of it has a monochromatic Schur triple, i.e., a triple of the form (a, b, a + b). This result is optimal since there are dense sets A, for which A ∩ [n]_p does not possess this property for p = o(n^-2/3).

Original language	English
Pages (from-to)	2175-2180
Number of pages	6
Journal	SIAM Journal on Discrete Mathematics
Volume	33
Issue number	4
DOIs	https://doi.org/10.1137/18M1227007
State	Published - 2019

Keywords

Ramsey theory
Random sets
Schur triples

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10.1137/18M1227007

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title = "Monochromatic Schur triples in randomly perturbed dense sets of integers",

abstract = "Given a dense subset A of the first n positive integers, we provide a short proof showing that for p = ω(n-2/3), the so-called randomly perturbed set A∩ [n]p a.a.s. has the property that any 2-coloring of it has a monochromatic Schur triple, i.e., a triple of the form (a, b, a + b). This result is optimal since there are dense sets A, for which A ∩ [n]p does not possess this property for p = o(n-2/3).",

keywords = "Ramsey theory, Random sets, Schur triples",

author = "Elad Aigner-Horev and Yury Person",

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year = "2019",

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AU - Aigner-Horev, Elad

AU - Person, Yury

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N2 - Given a dense subset A of the first n positive integers, we provide a short proof showing that for p = ω(n-2/3), the so-called randomly perturbed set A∩ [n]p a.a.s. has the property that any 2-coloring of it has a monochromatic Schur triple, i.e., a triple of the form (a, b, a + b). This result is optimal since there are dense sets A, for which A ∩ [n]p does not possess this property for p = o(n-2/3).

AB - Given a dense subset A of the first n positive integers, we provide a short proof showing that for p = ω(n-2/3), the so-called randomly perturbed set A∩ [n]p a.a.s. has the property that any 2-coloring of it has a monochromatic Schur triple, i.e., a triple of the form (a, b, a + b). This result is optimal since there are dense sets A, for which A ∩ [n]p does not possess this property for p = o(n-2/3).

KW - Ramsey theory

KW - Random sets

KW - Schur triples

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JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 4

ER -

Monochromatic Schur triples in randomly perturbed dense sets of integers

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Keywords

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