On a Conjecture of Frankl and Füredi

Abstract

Frankl and Füredi conjectured that if ${\cal F} \subset 2^{X}$ is a non-trivial $\lambda$-intersecting family of size $m$, then the number of pairs $\{x,y\} \in \binom{X}{2}$ that are contained in some $F \in {\cal F}$ is at least $\binom{m}{2}$ [P. Frankl and Z. Füredi. A Sharpening of Fisher's Inequality. Discrete Math., 90(1):103-107, 1991]. We verify this conjecture in some special cases, focusing especially on the case where ${\cal F}$ is additionally required to be $k$-uniform and $\lambda$ is small.