On Turan's theorem for sparse graphs
type: Item
instance of: scholarly article
title: On Turan's theorem for sparse graphs
zbMATH Open document ID: 0491.05038
DOI: 10.1007/BF02579451
author: Miklós Ajtai
author: János Komlós
author: P. Erdős
author: Endre Szemerédi
published in: Combinatorica
publication date: 1981-01-01T00:00:00Z
review text: Let \(\alpha\) be the (vertex) independence number and let \(\log x=\max\{1,\ell nx\}\). Denote by \(f(n,t,p)\) the largest integer such that every graph of order \(n\) and average degree \(t\geq 1\) that contains no \(K_p\) satisfies \(\alpha\geq f(n,t,p)\). Theorem: There exists an absolute constant \(c_1\) such that \(f(n,t,p)>c_1\cdot(n/t)\cdot\log(\log t)/p\). This improves on the known bounds \(\alpha\geq n/(t+1)\) and \(\alpha>0.01(n/t)\log t\). The lasr inequality may be rewritten as \(f(n,t,3)>c\cdot(n/t)\log t\), and suggests the study of the question \(f(n,t,p)=c_p(n/t)\log t\).
Mathematics Subject Classification ID: 05C35
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 3771639
zbMATH Keywords: independent set
zbMATH Keywords: clique
zbMATH Keywords: random graphs
Wikidata QID: Q105659312
reviewed by: S. F. Kapoor
MaRDI profile type: Publication
cites work: A note on Ramsey numbers
cites work: Turan's theorem for \(k\)-graphs
full work available at URL: https://doi.org/10.1007/bf02579451
OpenAlex ID: W2124088817
description: scientific article; zbMATH DE number 3771639