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**Zentralblatt MATH**

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

**Zbl.No: ** 199.31801

**Autor: ** Erdös, Pál

**Title: ** On a combinatorial problem. III (In English)

**Source: ** Can. Math. Bull. 12, 413-416 (1969).

**Review: ** [Part I in Nordisk. Mat. Tidskr. 11, 5-10 (1963; Zbl 116.01104)]

*E.W.Miller*, in C. R. Soc. Sci. Varsovie 30, 31-38 (1937; Zbl 017.30003), defines a family of sets **{**A**}** th have property B if there exists a set S which meets all the sets A_{k} and contains none of them. The author and *A.Hajnal* [Acta Math. Acad. Sci. Hungar. 12, 87-123 (1961; Zbl 201.32801)] define m(n) as the smallest integer for which there is a family of m(n) sets, each with cardinality n, which do not have property B. In Part II [ibid. 15, 445-447 (1964)], the author had found bounds for m(n). In this paper he considers the function m_{N}(n) which is the smallest integer for which there are m_{N}(n) sets A_{k} each with cardinality n which are all subsets of a set S, |S| = N, and which do not have property B. It is shown that if N = (c+o(1))n then **lim**_{n ––> oo} m_{N} (n)^{1/n} = 2(c-2)^{(c-2)/2}(c-1)^{(1-c)}c^{c/2}, if c > 2 and = 4 if c = 2. To prove this, upper and lower bounds for m_{N}(n) are found, differing byonly 2N. The author suggests that for large values of N the more appropriate function to consider would be m_{N}(n) being the smallest integer for which there is a family of sets not having property B, satisfying A_{k} \subset S, |S| = N with the restriction that the set of A_{k}'s contained in any proper subset of S has the property B. A symptotic formulae for m_{N}(n) and m_{N}'(n) are not known.

**Reviewer: ** D.I.A.Cohen

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

04A99 Miscellaneous topics in set theory

**Index Words: ** combinatorics

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