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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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##### References:
 [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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