On a Conjecture of Frankl and Füredi

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