论文信息 - Subsets of a Finite Set That Intersect Each Other in At Most One Element

Subsets of a Finite Set That Intersect Each Other in At Most One Element

Abstract In this paper we study de Bruijn-Erdos type theorems that deal with the foundations of finite geometries. The following theorem is one of our main conclusions. Let S1,…, Sn be n subsets of an n-set S. Suppose that |Si| ⩾ 3 (i = 1,…,n) and that |Si ∩ Sj| ⩽ 1 (i ≠ j;i,j = 1,…,n). Suppose further that each Si has nonempty intersection with at least n − 2 of the other subsets. Then the subsets S1,…,Sn of S are one of the following configurations. (1) They are a finite projective plane. (2) They are a symmetric group divisible design and each subset has nonempty intersection with exactly n − 2 of the other subsets. (3) We have n = 9 or n = 10 and in each case there exists a unique configuration that does not satisfy (1) or (2).

H.J Ryser | H. Ryser

[1] de Ng Dick Bruijn. A combinatorial problem , 1946 .

[2] H. J. Ryser. Variants of cyclic difference sets , 1973 .

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

[4] R. C. Bose,et al. On the construction of group divisible incomplete block designs , 1953 .

[5] Douglas R. Woodall. Square λ-Linked Designs , 1970 .

[6] On the number of absolute points of a correlation. , 1956 .

[7] H. J. Ryser. Symmetric Designs and Related Configurations , 1972, J. Comb. Theory, Ser. A.

[8] Stanley E. Payne,et al. On the Non-Existence of a Class of Configurations Which Are Nearly Generalized n-Gons , 1972, J. Comb. Theory, Ser. A.

[9] W. S. Connor,et al. Combinatorial Properties of Group Divisible Incomplete Block Designs , 1952 .

[10] J. Gillis,et al. Matrix Iterative Analysis , 1961 .

[11] O. Taussky. A Recurring Theorem on Determinants , 1949 .