Intersecting families with covering number five

A family F subset of (([n])(k)) is called intersecting if any two members of it have non-empty intersection. The covering number of F is defined as the minimum integer p such that there exists T subset of {1,2,& mldr;,n} satisfying |T| = p and T boolean AND F not equal & empty; for all F is an element of F. Define m(n, k, p) as the maximum size of an intersecting family F subset of (([n])(k)) with covering number at least p. The value of m(n, k, p) is only known for p = 1,2,3,4. About thirty years ago, m(n, k, 5) was determined asymptotically by the first author, Ota and Tokushige. In the present paper, we determine m(n, k, 5) for k >= 69 and n >= 5k(6). (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, Al training, and similar technologies