Intersection properties of systems of finite sets

M. Deza, P. Erdős, P. Frankl

Research output: Contribution to journalArticle

46 Citations (Scopus)

Abstract

Let X be a finite set of cardinality n. If L = (l1,…, lr) is a set of nonnegative integers with l12<… <lr, and & is a natural number, then by an (n, L, k)-system we mean a collection of k-element subsets of X such that the intersection of any two different sets has cardinality belonging to L. We prove that if A is an (n, L, k)-system, with A > cnr-1 (c = c(k) is a constant depending on k), then (i) there exists an l1-element subset D of X such that D is contained in every member of A (ii) (l2-l1)(l3-l2)… (iii) (Πri=1 (n-li)/(k-li) ≥ A (for n ≥ n0(k)). Parts of the results are generalized for the following cases: (a) we consider t-wise intersections, where t ≥ 2; (b) the condition A= k is replaced by A K where K is a set of integers; (c) the intersection condition is replaced by the following: Among q +1 different members A1,…, Aq+1 there are always two, Ai, Aj, such that e Ai ∩ Aj ∈ L. We consider some related problems. An open question: let L’ ⊂ L do there exist an [n, L, k)-system of maximal cardinality (A) and an (n, L’, k)-system of maximal cardinality (A’) such that A A’?.

Original languageEnglish
Pages (from-to)369-384
Number of pages16
JournalProceedings of the London Mathematical Society
Volumes3-36
Issue number2
DOIs
Publication statusPublished - 1978

ASJC Scopus subject areas

  • Mathematics(all)

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