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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 384.05001

**Autor: ** Deza, M.; Erdös, Paul; Frankl, P.

**Title: ** Intersection properties of systems of finite sets. (In English)

**Source: ** Combinatorics, Keszthely 1976, Colloq. Math. Janos Bolyai 18, 251-256 (1978).

**Review: ** [For the entire collection see Zbl 378.00007.]

Let X be a finite set of cardinality n. If L = **{**l_{1},...,\l_{r}**}** is a set of nonnegative integers l_{1} < l_{2} < ... < l_{r}, and k is a natural number then by an (n,L,k)-system we mean a collection of k-element subsets of X such that the interesection of any two different sets has cardinality belonging to L. We prove that if *A* is an (n,L,k)-system, |*A*| > cn^{r-1} (c = c(k) is a constant depending on k) then (i) there exists an l_{1}-element subset D of X such that D is contained in every member of *A*, (ii) (l_{2}-l_{1})|/l_{3}-l_{2})|...|(l_{r}-l_{r-1})|(k-l_{r}), (iii) **prod**_{i = 1}^{r}\frac{n-l_{i}}{k-l_{i}} \geq | *A*| for n \geq n_{0}(k)).

Parts of the results are generalized for the following cases: (a) we consider t-wise intersections, t \geq 2, (b) the condition |A| = k is replaced by |A| in K where K is a set of integers, (c) the intersection condition is replaced by the following: among q+1 different members A_{1},...,A_{q+1} there are always two subsets A_{i}, A_{j} such that |A_{i}\cap A_{j}| in L. We consider some related problems. An open question: Let L'\subset L. Does there exist an (n,L,k)-system of maximal cardinality (*A*) and an (n,L',k)-system of maximal cardinality (*A*) such that *A*\supset *A*'?

**Classif.: ** * 05A05 Combinatorial choice problems

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