An improved lower bound for Folkman's theorem

József Balogh; Sean Eberhard; Bhargav Narayanan; Andrew Treglown; Adam Zsolt Wagner

doi:10.1112/blms.12058

An improved lower bound for Folkman's theorem

József Balogh^*
, Sean Eberhard
, Bhargav Narayanan
, Andrew Treglown
, Adam Zsolt Wagner

^*Corresponding author for this work

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

2 Citations (Scopus)

Abstract

Folkman's theorem asserts that for each kϵN, there exists a natural number n=F(k) such that whenever the elements of [n] are two-coloured, there exists a set A C[n] of size k with the property that all the sums of the form σ_xϵB^x, where B is a nonempty subset of A, are contained in [n] and have the same colour. In 1989, Erds and Spencer showed that F(k)≥2ck2/logk, where c>0 is an absolute constant; here, we improve this bound significantly by showing that F(k)≥22k-1/k for all kϵN.

Original language	English
Pages (from-to)	745-747
Number of pages	3
Journal	Bulletin of the London Mathematical Society
Volume	49
Issue number	4
DOIs	https://doi.org/10.1112/blms.12058
Publication status	Published - Aug 2017
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2017 London Mathematical Society.

Keywords

05D10 (primary)
05D40 (secondary)

ASJC Scopus subject areas

General Mathematics

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10.1112/blms.12058

Cite this

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title = "An improved lower bound for Folkman's theorem",

abstract = "Folkman's theorem asserts that for each kϵN, there exists a natural number n=F(k) such that whenever the elements of [n] are two-coloured, there exists a set A C[n] of size k with the property that all the sums of the form σxϵBx, where B is a nonempty subset of A, are contained in [n] and have the same colour. In 1989, Erds and Spencer showed that F(k)≥2ck2/logk, where c>0 is an absolute constant; here, we improve this bound significantly by showing that F(k)≥22k-1/k for all kϵN.",

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T1 - An improved lower bound for Folkman's theorem

AU - Balogh, József

AU - Eberhard, Sean

AU - Narayanan, Bhargav

AU - Treglown, Andrew

AU - Wagner, Adam Zsolt

PY - 2017/8

Y1 - 2017/8

N2 - Folkman's theorem asserts that for each kϵN, there exists a natural number n=F(k) such that whenever the elements of [n] are two-coloured, there exists a set A C[n] of size k with the property that all the sums of the form σxϵBx, where B is a nonempty subset of A, are contained in [n] and have the same colour. In 1989, Erds and Spencer showed that F(k)≥2ck2/logk, where c>0 is an absolute constant; here, we improve this bound significantly by showing that F(k)≥22k-1/k for all kϵN.

AB - Folkman's theorem asserts that for each kϵN, there exists a natural number n=F(k) such that whenever the elements of [n] are two-coloured, there exists a set A C[n] of size k with the property that all the sums of the form σxϵBx, where B is a nonempty subset of A, are contained in [n] and have the same colour. In 1989, Erds and Spencer showed that F(k)≥2ck2/logk, where c>0 is an absolute constant; here, we improve this bound significantly by showing that F(k)≥22k-1/k for all kϵN.

KW - 05D10 (primary)

KW - 05D40 (secondary)

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M3 - Article

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JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

IS - 4

ER -

An improved lower bound for Folkman's theorem

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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