Intersection Properties of Systems of Finite Sets

Let X be a finite set of cardinality n. If L = {ll , . . ., l,} is a set of nonnegative integers with 1 1 < 1 2 < . . . < lr , and k is a natural number, then by an (n, L, k)-system we mean a collection of k-element subsets of X such that the intersection of any two different sets has cardinality belonging to L. We prove that if d is an (n, L, k)-system, with I ~V I > enr -1 (c = e(k) is a constant depending on k), then (i) there exists an l1-element subset D of X such that D is contained in every member of Q, (11) (12 11)1(13 12)1 . . .I(lr lr-1)I (k-l r), (iii) ni=1 (?,-li)/(kli) > ICI (for n > n,(k)) . Parts of the results are generalized for the following cases : (a) we consider t-wise intersections, where t > 2 ; (b) the condition I A I = k is replaced by I A I E K where K is a set of integers ; (c) the intersection condition is replaced by the following : among q + 1 different members A 1 , . . ., Aq+1 there are always two, A i ,Aj , such that I A i n A, I E L. We consider some related problems . An open question : let L' L ; do there exist an (n, L, k)-system of maximal cardinality (s2/) and an (n, L', k)-system of maximal cardinality (,') such that d ::) a'? 1 . Introduction Throughout this paper lower case latin letters denote integers, capital letters stand for sets, and capital script letters for families of sets . Let L = {l l , . . ., l,}, where 1 < 12 < . . . < l r , and K be sets of integers . By an (n, L, K)-system we mean a family Q of subsets of a set X, with I X I = n, such that for A t , A 2 E d we have IA1 1, A 2 I E K, I A 1 n A 2 I E L. If K = {k} then the notation (n, L, k)-system is applied, too . A family B = {B1, B 2 , . . ., B.} of sets is called a 0-system of cardinality c if there exists a set D Bi , with i = 1, . . ., c, such that the sets B1\D, . . .,B,\D are pairwise disjoint . D is called the kernel of the 0-system . Proc. London Math . Soc. (3) 36 (1978) 369-384