A new generalization of the Erdős-Ko-Rado theorem

Peter Frankl, Zoltán Füredi

Research output: Contribution to journalArticle

45 Citations (Scopus)

Abstract

Let ℱ be a family of k-subsets of an n-set. Let s be a fixed integer satisfying k≦s≦3 k. Suppose that for F 1, F 2, F 3 ∈ ℱ |F 1 ∪F 2 ∪F 3|≦s implies F 1 ∩F 2 ∩F 3 ≠ 0. Katona asked what is the maximum cardinality, f(n, k, s) of such a system. The Erdo{combining double acute accent}s-Ko-Rado theorem implies f(n, k, s)= {Mathematical expression} for s=3 k and n≧2 k. In this paper we show that f(n, k, s)= {Mathematical expression} holds for n>n 0(k) if and only if s≧2 k. Equality holds only if every member of ℱ contains a fixed element of the underlying set. Further we solve the problem for k=3, s=5, n≧3000. This result sharpens a theorem of Bollobás.

Original languageEnglish
Pages (from-to)341-349
Number of pages9
JournalCombinatorica
Volume3
Issue number3-4
DOIs
Publication statusPublished - Sep 1983

    Fingerprint

Keywords

  • AMS subject classification (1980): 05C35, 05B30

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

Cite this