Two-Point Concentration of the Independence Number of the Random Graph

IF 1.2 2区数学 Q1 MATHEMATICS Forum of Mathematics Sigma Pub Date : 2024-02-23 DOI:10.1017/fms.2024.6

Tom Bohman, Jakob Hofstad

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引用次数: 0

Abstract

We show that the independence number of

$ G_{n,p}$

is concentrated on two values if

$ n^{-2/3+ \epsilon } < p \le 1$

. This result is roughly best possible as an argument of Sah and Sawhney shows that the independence number is not, in general, concentrated on two values for

$ p = o ( (\log (n)/n)^{2/3} )$

. The extent of concentration of the independence number of

$ G_{n,p}$

for

$ \omega (1/n) < p \le n^{-2/3}$

remains an interesting open question.

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随机图独立数的两点浓度

我们证明，如果 $ n^{-2/3+ \epsilon } < p \le 1$，$ G_{n,p}$ 的独立数会集中在两个值上。Sah 和 Sawhney 的论证表明，在一般情况下，当 $ p = o ( (\log (n)/n)^{2/3} )$ 时，独立数不会集中在两个值上。在 $ \omega (1/n) < p \le n^{-2/3}$ 时，$ G_{n,p}$ 的独立数的集中程度仍然是一个有趣的未决问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Forum of Mathematics Sigma Mathematics-Statistics and Probability

CiteScore

1.90

自引率

5.90%

发文量

审稿时长

40 weeks

期刊介绍： Forum of Mathematics, Sigma is the open access alternative to the leading specialist mathematics journals. Editorial decisions are made by dedicated clusters of editors concentrated in the following areas: foundations of mathematics, discrete mathematics, algebra, number theory, algebraic and complex geometry, differential geometry and geometric analysis, topology, analysis, probability, differential equations, computational mathematics, applied analysis, mathematical physics, and theoretical computer science. This classification exists to aid the peer review process. Contributions which do not neatly fit within these categories are still welcome. Forum of Mathematics, Pi and Forum of Mathematics, Sigma are an exciting new development in journal publishing. Together they offer fully open access publication combined with peer-review standards set by an international editorial board of the highest calibre, and all backed by Cambridge University Press and our commitment to quality. Strong research papers from all parts of pure mathematics and related areas will be welcomed. All published papers will be free online to readers in perpetuity.

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