For most graphs H, most H‐free graphs have a linear homogeneous set1. Issue 3 (18th March 2013)

• Record Type: Journal Article
• Authors: Kang, Ross J.; McDiarmid, Colin; Reed, Bruce; Scott, Alex
• Is Part Of: Random structures & algorithms. Volume 45:Issue 3(2014)
• Page Start: 343
• Page End: 361
• Publication Date: 2013-03-18
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20488
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>ABSTRACT</title> <p>Erdős and Hajnal conjectured that for every graph <italic>H</italic> there is a constant <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v95fmv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20488:rsa20488-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ε</mml:mo><mml:mo>=</mml:mo><mml:mo>ε</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> such that every graph <italic>G</italic> that does not have <italic>H</italic> as an induced subgraph contains a clique or a stable set of order <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v95f0x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20488:rsa20488-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Ω</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>|</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mo>|</mml:mo><mml:mo>ε</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>.</p> <p>The conjecture would be false if we set <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v95f1g" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20488:rsa20488-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ε</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>; however, in an asymptotic setting, we obtain this strengthened form of Erdős and Hajnal's conjecture for almost every graph <italic>H</italic>, and in particular for a large class of graphs <italic>H</italic> defined by variants of the colouring number. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 343–361, 2014</p> </abstract>