Intersection theorems with geometric consequences
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In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ0, μ1, ..., μs are distinct residues modp (p is a prime) such thatk ≡ μ0 (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μi (modp) for somei, 1 ≦i≦s, then |ℱ|≦( s n ).
As a consequence we show that ifR n is covered bym sets withm<(1+o(1)) (1.2) n then there is one set within which all the distances are realised.
It is left open whether the same conclusion holds for compositep.
AMS subject classification (1980)05 C 65 05 C 35 05 C 15
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- M. Deza, P. Erdős andN. M. Singhi, Combinatorial problems on subsets and their intersections,Advances in Mathematics, Suppl. Studies1 (1978), 259–265.Google Scholar
- P. Erdős, Problems and results in graph theory and combinatorial analysis,Proc. Fifth British Comb. Conf. 1975 Aberdeen, Congressus Numerantium,15 — Utilitas Mathematica, Winnipeg, 1976.Google Scholar
- P. Erdős, Problems and results in chromatic graph theory, in:Proof techniques in graph theory (F. Harary ed.), Academic Press, London, 1969, 27–35.Google Scholar
- P. Frankl, Problem session,Proc. French—Canadian Joint Comb. Coll., Montreal 1978.Google Scholar
- P. Frankl, Families of finite sets with prescribed cardinalities for pairwise intersections,Acta Math. Acad. Sci. Hung., to appear.Google Scholar
- P. Frankl andI. G. Rosenberg, An intersection problem for finite sets, Europ. J. Comb2 (1981).Google Scholar