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Combinatorial properties of systems of sets. (English) Zbl 0383.05002
A 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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##### References:
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