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Regularly intersecting families of finite sets. (English) Zbl 0608.05011

If p is a prime, \({\mathcal F}_ i\) \((i=1,...,t)\) is a family of subsets of a set having n elements, then \({\mathfrak F}:=\{{\mathcal F}_ 1,...,{\mathcal F}_ t\}\) is called p-regularly intersecting if there exist integers \(x_ i\) and \(y_{ij}\) such that \(x_ i\not\equiv y_{ii}\), \(| F'| \equiv x_ i\), \(| F'\cap F''| \equiv y_{ij}(mod p)\) for all \(1\leq i,j\leq t\), F’\(\in {\mathcal F}_ i\), F”\(\in {\mathcal F}_ j\) with F’\(\neq F''\) if \(i=j\). The main result is that \(m:=| {\mathcal F}_ 1| +...+| {\mathcal F}_ t| <n+t\) (Theorem 10). In order to prove this, the author establishes some properties of the vectors with prescribed scalar products.
Reviewer: M.Ştefănescu

MSC:

05A05 Permutations, words, matrices
05B05 Combinatorial aspects of block designs
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