论文信息 - Intersection theorems with geometric consequences

Intersection theorems with geometric consequences

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|≦(sn).As a consequence we show that ifRn 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.

Peter Frankl | Richard M. Wilson | P. Frankl | R. Wilson

[1] H. Hadwiger,et al. Ein ?berdeckungssätze für den Euklidischen Raum , 1944 .

[2] Peter Frankl. An intersection problem for finite sets , 1977 .

[3] H. Hadwiger. Ueberdeckung des Euklidischen Raumes durch kongruente Mengen , 1945 .

[4] Peter Frankl. Extremal Problems and Coverings of the Space , 1980, Eur. J. Comb..

[5] Peter Frankl,et al. On Set Intersections , 1980, J. Comb. Theory, Ser. A.

[6] Peter Frankl,et al. A Finite Set Intersection Theorem , 1981, Eur. J. Comb..

[7] H. Ryser,et al. An extension of a theorem of de Bruijn and Erdös on combinatorial designs , 1968 .

[8] P. Erdös. Some remarks on the theory of graphs , 1947 .

[9] Peter Frankl. Families of finite sets with prescribed cardinalities for pairwise intersections , 1980 .

[10] C. Rogers,et al. The realization of distances within sets in Euclidean space , 1972 .

[11] D. G. Larman. A note on the realization of distances within sets in euclidean space , 1978 .

[12] Paul Erdös,et al. INTERSECTION PROPERTIES OF SYSTEMS OF FINITE SETS , 1978 .

[13] Richard M. Wilson,et al. On $t$-designs , 1975 .