On the Representing Number of Intersecting Families

1. Introduction. One of the best-known results in extremal set theory is the Theorem of Erdös-Ko-Rado [3] : Suppose n > 2 k, and let 9i be a family of k-subsets of an n-set M such that any two members of M intersect non-trivially, then I J1I < (k-1). Furthermore, the bound can be attained, and the extremal families are precisely the families SJ7ta = {X D a : a E M{ for k > 3. Many proofs of this result have been given, in addition to the original proof see e. g. [4, 9, 10]. Since all the members of an extremal familiy J2 have an element in common, we say that M has representing number 1. What if we do not allow the sets of Ti to have an overall nontrivial intersection? How large can then V be? The answer to this question has been given by Hilton-Milner [8] with a further proof appearing e. g. in [6] : Let 9Y be an intersecting family of k-subsets of an n-set M such that X = ¢ then < n-1 n-k-1 n I_ k-1 k-1)+1 for n>2k. Xem Again the extremal families are characterized. Since the members of 9t are allowed to contain one of two points, but not a single one we say that 9t has representing number 2. In this paper we estimate the cardinality of an intersecting family with an arbitrary representing number r, 1 <_ r _< k. We first give the relevant definitions. All sets will be assumed to be finite. The collection of all k-subsets of a set M will be denoted by CM). We say that a family M is intersecting if any two members of M have a non-trivial intersection .