Uniform Intersecting Families with Covering Number Restrictions

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, p_t(k) ≤ k^t - 1/√2 ⌊t-1/2⌋^3/2k^t-1 + O(k^t-2). In the cases of t = 4 and 5, we also prove that the coefficient of k^t-1 in p_t(k) is exactly (^t₂).