, Volume 1, Issue 4, pp 357–368 | Cite as

Intersection theorems with geometric consequences

  • P. Frankl
  • R. M. Wilson


In this paper we prove that if 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′ we have |FF′| ≡ μi (modp) for somei, 1 ≦is, then ||≦( 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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Copyright information

© Akadémiai Kiadó 1981

Authors and Affiliations

  • P. Frankl
    • 1
  • R. M. Wilson
    • 2
  1. 1.C. N. R. S.ParisFrance
  2. 2.California Inst. TechnologyPasadenaU.S.A.

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