Mantel's theorem for random graphs1. Issue 1 (28th March 2014)

• Record Type: Journal Article
• Authors: DeMarco, B.; Kahn, J.
• Is Part Of: Random structures & algorithms. Volume 47:Issue 1(2015)
• Page Start: 59
• Page End: 72
• Publication Date: 2014-03-28
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20535
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>ABSTRACT</title> <p>For a graph <italic>G</italic>, denote by <italic>t</italic>(<italic>G</italic>) (resp. <italic>b</italic>(<italic>G</italic>)) the maximum size of a triangle‐free (resp. bipartite) subgraph of <italic>G</italic>. Of course <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj14cqk94h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20535:rsa20535-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≥</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> for any <italic>G</italic>, and a classic result of Mantel from 1907 (the first case of Turán's Theorem) says that equality holds for complete graphs. A natural question, first considered by Babai, Simonovits and Spencer about 20 years ago is, when (i.e., for what <italic>p</italic> = <italic>p</italic>(<italic>n</italic>)) is the "Erdős‐Rényi" random graph <italic>G</italic> = <italic>G</italic>(<italic>n</italic>, <italic>p</italic>) likely to satisfy <italic>t</italic>(<italic>G</italic>) = <italic>b</italic>(<italic>G</italic>)? We show that this is true if <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj14cqk91v" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20535:rsa20535-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>></mml:mo><mml:mi>C</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives></inline-formula> for a suitable constant <italic>C</italic>, which is best possible up to the value of <italic>C</italic>. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 59–72, 2015</p> </abstract>