Intersection Properties of Subsets of Integers

Miklós Simonovits, Vera T. Sós

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Abstract

Let {A1,...,AN} be a family of subsets of {1, 2,...,n}. For a fixed integer k we assume that keyable*** is an arithmetic progression of ⩾k elements for every 1 ⩽ i < j ⩽ N. We would like to determine the maximum of N. For k = 0, R. L. Graham and the authors have proved that N=⩽ ( n3)+ ( n2)+ ( n1)+1 For k ⩾ 2, the extremal and asymptotically extremal systems have ( π2 24+o(1))n2sets For k = 1, the maximum is between ( n2)+1and ( π2 24+12o(1))n2 We conjecture that the lower bound is sharp.

Original languageEnglish
Pages (from-to)363-372
Number of pages10
JournalEuropean Journal of Combinatorics
Volume2
Issue number4
DOIs
Publication statusPublished - Jan 1 1981

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ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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