Covers in 4-uniform Intersecting Families with Covering Number Three

Abstract

Let $k$ be an integer. In [3, 4], Frankl, Ota and Tokushige proved that the maximum number of three-covers of a $k$-uniform intersecting family with covering number three is $k^3 - 3k^2 + 6k -4$ for $k=3$ or $k \ge 9$, but the case $4 \le k \le 8$ remained open. In this paper, we prove that the same holds for $k=4$, and show that a 4-uniform family with covering number three which has 36 three-covers is uniquely determined.