Uniform Intersecting Families with Covering Number Restrictions

Source abstract

It is known that any k -uniform family with covering number t has at most k t t -covers. In this paper, we deal with intersecting families and give better upper bounds for the number of t -covers. Let p t ( k ) be the maximum number of t -covers in any k -uniform intersecting families with covering number t . We prove that, for a fixed t , formula here In the cases of t =4 and 5, we also prove that the coefficient of k t −1 in p t ( k ) is exactly ( t 2 ).