Intersecting families, cross-intersecting families, and a proof of a conjecture of Feghali, Johnson and Thomas

Abstract

A family A of sets is said to be intersecting if every two sets in A intersect. Two families A and B are said to be cross-intersecting if each set in A intersects each set in B. For a positive integer n, let [n]={1,…,n} and Sn={A⊆[n]:1∈A}. We extend the Erdős–Ko–Rado Theorem by showing that if A and B are non-empty cross-intersecting families of subsets of [n], A is intersecting, and a0,a1,…,an,b0,b1,…,bn are non-negative real numbers such that ai+bi≥an−i+bn−i and an−i≥bi for each i≤n∕2, then ∑A∈Aa|A|+∑B∈Bb|B|≤∑A∈Sna|A|+∑B∈Snb|B|.For a graph G and an integer r≥1, let IG(r) denote the family of r-element independent sets of G. Inspired by a problem of Holroyd and Talbot, Feghali, Johnson and Thomas conjectured that if r<n and G is a depth-two claw with n leaves, then G has a vertex v such that {A∈IG(r):v∈A} is a largest intersecting subfamily of IG(r). They proved this for r≤n+12. We use the result above to prove the full conjecture.