On a conjecture of Frankl and Füredi (Q540036)

Summary: Frankl and Füredi conjectured that if \(\mathcal F \subset 2^X\) 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 2 some \(F \in \mathcal F\) is at least \(m\) [\textit{P. Frankl} and \textit{Z. Füredi}, ``A Sharpening of Fisher's 2 Inequality''. Discrete Math. 90, No. 1, 103--107 (1991; Zbl 0762.05083)]. We verify this conjecture in some special cases, focusing especially on the case where \(\mathcal F\) is additionally required to be \(k\)-uniform and \(\lambda\) is small.