A topological colorful Helly theorem

Let F₁,...,F_d+1 be d + 1 families of convex sets in ℝ^d. The Colorful Helly Theorem (see (Discrete Math. 40 (1982) 141)) asserts that if ∩_i=1^d+1 F_i≠∅ for all choices of F₁∈ F₁,...,F_d+1∈ F_d+1 then there exists an 1≤i≤d+1 such that ∩_F∈Fi F≠∅. Our main result is both a topological and a matroidal extension of the colorful Helly theorem. A simplicial complex X is d-Leray if H_i(Y;ℚ)=0 for all induced subcomplexes Y⊂X and i≥d. Theorem. Let X be a d-Leray complex on the vartex set V. Suppose M is a matroidal complex on the same vertex set V with rank function ρ. If M⊂X then there exists a simplex τ∈X such that ρ(V-τ)≤d.