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Sharpness of some intersection theorems. (English) Zbl 0971.05109
A theorem of P. Frankl and I. G. Rosenberg [A finite set intersection theorem, Eur. J. Comb. 2, 127-129 (1981; Zbl 0461.05001)] states: Let \(0\leq \mu_1< p\) be integers and let \({\mathcal F}\) be a family of \(k\)-element subsets of an \(n\)-element set such that \(k\not\equiv \mu_1\pmod p\) and \(|F\cap F'|\equiv \mu_1\pmod p\) for any two distinct members \(F\), \(F'\) of \({\mathcal F}\). Then \(|{\mathcal F}|\leq n\).
The authors construct infinite sequences of non-trivial families for which the bound is attained. Some further constructions are given for related inequalities.
MSC:
05D05 Extremal set theory
05B05 Combinatorial aspects of block designs
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[1] Deza, M.; Rosenberg, I., Cardinalités de sommets et d’arêtes d’hypergraphes satisfaisant à certaines conditions sur l’intersection d’arêtes, Cahiers cent. étud. rech. opér., 20, 279-285, (1978) · Zbl 0429.05056
[2] Frankl, P.; Rosenberg, I.G., A finite set intersection theorem, Europ. J. combinatorics, 2, 127-129, (1981) · Zbl 0461.05001
[3] Frankl, P.; Rosenberg, I.G., Regularly intersecting families of finite sets, Ars comb., 22, 97-105, (1986) · Zbl 0608.05011
[4] Frankl, P.; Wilson, R.M., Intersection theorems with geometric consequences, Combinatorica, 1, 357-368, (1981) · Zbl 0498.05048
[5] Knuth, D.E., The art of computer programming. 1: fundamental algorithms, (1979), Addison-Wesley Reading, MA · Zbl 0191.17903
[6] Wilson, R.M., An existence theory for pairwise designs III: proof of the existence conjectures, J. comb. theory, ser. A, 18, 71-79, (1975) · Zbl 0295.05002
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