Intersection properties of systems of finite sets

M. Deza, P. Erdős, P. Frankl

Research output: Contribution to journalArticle

45 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 language English 369-384 16 Proceedings of the London Mathematical Society s3-36 2 https://doi.org/10.1112/plms/s3-36.2.369 Published - 1978

Finite Set
Cardinality
Intersection
Integer
Non-negative
Subset

ASJC Scopus subject areas

• Mathematics(all)

Cite this

Intersection properties of systems of finite sets. / Deza, M.; Erdős, P.; Frankl, P.

In: Proceedings of the London Mathematical Society, Vol. s3-36, No. 2, 1978, p. 369-384.

Research output: Contribution to journalArticle

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