It is known that any k-uniform family with covering number t has at most kt t-covers. In this paper, we deal with intersecting families and give better upper bounds for the number of t-covers. Let pt(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, pt(k) ≤ kt - 1/√2 ⌊t-1/2⌋3/2kt-1 + O(kt-2). In the cases of t = 4 and 5, we also prove that the coefficient of kt-1 in pt(k) is exactly (t2).
ASJC Scopus subject areas
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics