Note on a problem of M. Talagrand1. Issue 4 (1st July 2014)

• Record Type: Journal Article
• Authors: DeMarco, B.; Kahn, J.
• Is Part Of: Random structures & algorithms. Volume 47:Issue 4(2015)
• Page Start: 663
• Page End: 668
• Publication Date: 2014-07-01
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20559
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Answering a question of M. Talagrand, we show that there is a fixed <italic>L</italic> with the following property. For positive integers <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgc4q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20559:rsa20559-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>≤</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgc2p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20559:rsa20559-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:mo stretchy="false">[</mml:mo><mml:mn>0</mml:mn><mml:mo>, </mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, if <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgc0n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20559:rsa20559-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>ℱ</mml:mo></mml:math></alternatives></inline-formula> is the set of subgraphs of <italic>K</italic><sub><italic>n</italic></sub> containing at least <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgbwm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20559:rsa20559-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mtable><mml:mtr><mml:mtd><mml:mi>n</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>k</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mtable><mml:mtr><mml:mtd><mml:mi>k</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>2</mml:mn></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula> copies of <italic>K</italic><sub><italic>k</italic></sub>, then there is a set <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgbv3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20559:rsa20559-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">G</mml:mi></mml:math></alternatives></inline-formula> of subgraphs of <italic>K</italic><sub><italic>n</italic></sub> such that (i) each member of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgbtk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20559:rsa20559-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>ℱ</mml:mo></mml:math></alternatives></inline-formula> contains a member of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgbs2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20559:rsa20559-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">G</mml:mi></mml:math></alternatives></inline-formula> and (ii) <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgbrj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20559:rsa20559-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>S</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo>/</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>S</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>≤</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> (where <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgbq1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20559:rsa20559-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo><mml:mi>S</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> means number of edges). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 663–668, 2015</p> </abstract>