intersection properties of systems of finite sets - CiteSeerX
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intersection properties of systems of finite sets - CiteSeerX
set c(k, L) = max(& â ^ + 1, Z2 ... Further, if qi denotes the maximum number of sets Av ...,Aq< such that, for. 1
INTERSECTION PROPERTIES OF SYSTEMS OF FINITE SETS By M. DEZA, P . ERDOS, and P . FRANKL [Received 30 June 1976]
Abstract Let X be a finite set of cardinality n. If L = {lv ...,lr} is a set of nonnegative integers with lx < l2 < ... < lr, and k is a natural number, then by an (n, L, k)-system we mean a collection of ^-element subsets of X such that the intersection of any two different sets has cardinality belonging to L. We prove that if s& is an (n,L, &)-system, with \s&\> cnr~x (c = c(k) is a constant depending on k), then (i) there exists an Zx-element subset D of X such that D is contained in every member of stf, (ii) {h-h)\{h-h)\-Wr-lr-r)\{k-lr), (iii) UUin-h)/{k-lt) >\j*\(£orn> no(k)). Parts of the results are generalized for the following cases: (a) we consider £-wise intersections, where t ^ 2; (b) the condition \A\ = k is replaced by | A \ e K where K is a set of integers; (c) the intersection condition is replaced by the following: among q +1 different members Ax,..., Aq+1 there are always two, A^Ap such that l ^ n A J e L. We consider some related problems. An open question: let U
