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**Zentralblatt MATH**

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

**Zbl.No: ** 970.56262

**Autor: ** Deuber, W.A.; Erdös, Paul; Gunderson, D.S.; Kostochka, A.V.; Meyer, A.G.

**Title: ** Intersection statements for systems of sets. (In English)

**Source: ** J. Comb. Theory, Ser. A 79, No.1, 118-132, Art. No.TA962778 (1997).

**Review: ** A collection of sets is called a \Delta-system if any two sets have the same intersection. Let f(k,r) be the least integer such that any collelction of f(k,r) k-element sets contains a \Delta-system consisting of r sets. *P. Erdös* and *R. Rado* [J. Lond. Math. Soc. 44, 467-479 (1969; Zbl 172.29601)] proved that (r-1)^{k} < f(k,r) < k!(r-1)^{k} and conjectured that f(k,r) < C^{k} for some constant C. Erdös offered $1000 for a proof or disproof of this for r = 3.

The paper under review concerns a related problem. Let F(n,r) be the greatest integer such that there exists a collection of subsets of an n-element set which does not contain a \Delta-system consiting of r sets. *P. Erdös* and *E. Szemerédi* [J. Comb. Theory, Ser. A 24, 308-313 (1978; Zbl 383.05002)] showed that F(n,3) < 2^{n-\sqrt n/10} and F(n,r) > (1+c_{r})^{n}, where the constant c_{r} ––> 1 as r ––> oo. The authors provide new lower bounds for F(n,r) which are constructive and improve the previous best probabilistic results. They also prove a new upper bound. Moreover, for certain n it is shown that F(n,3) \geq 1.551^{n-2}.

**Reviewer: ** A.Vince (Gainesville)

**Classif.: ** * 05D05 Extremal set theory

**Keywords: ** \Delta-system

**Citations: ** Zbl 172.29601; Zbl 383.05002

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