, Volume 1, Issue 4, pp 357-368

Intersection theorems with geometric consequences

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Abstract

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.