Combinatorial properties of systems of sets.

*(English)*Zbl 0383.05002A family of sets is called a strong (weak) \(\triangle\) system if the (cardinality of the) intersection of any two of its members is the same. The paper contains remarks, considerations, conjectures and results on the following functions: \(f(n,r) =\) smallest integer for which any family of \(f(n,r)\) sets of size \(n\) contains a subfamily of \(r\) sets which forms a strong \(\triangle\) system; \(g(n,r)\) is defined similarly for weak \(\triangle\) systems; \(F(n,r) =\) largest integer so that there is a family of subsets of an \(n\)-set which does not contain a strong \(\triangle\) system of \(r\) elements; \(G(n,r)\) has the similar meaning for weak \(\triangle\) systems; \(F(n,r,k)\) and \(G(n,r,k)\) are defined similarly with the sole distinction that only \(k\)-subsets are considered. The existence is proved of families of subsets of an \(n\)-set not containing weak \(\triangle\) systems and having at least \(n^{\log n/4 \log\log n}\) members.

Reviewer: W. Dörfler

##### MSC:

05A05 | Permutations, words, matrices |

03E05 | Other combinatorial set theory |

11B75 | Other combinatorial number theory |

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\textit{P. Erdős} and \textit{E. Szemerédi}, J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)

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##### References:

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