Leray numbers of projections and a topological Helly-type theorem

Let X be a simplicial complex on the vertex set V. The rational Leray number L (X) of X is the minimal d, such that (Formula presented.) for all induced subcomplexes Y ⊂ X and i ⩾ d. Suppose that (Formula presented.) is a partition of V such that the induced subcomplexes X[V_i] are all 0-dimensional. Let π denote the projection of X into the (m − 1)-simplex on the vertex set {1, …, m} given by π(v) = i if v ∈ V_i. Let r = max{|π⁻¹(π(x))|:x ∈ |X|}. It is shown that (Formula presented.) One consequence is a topological extension of a Helly-type result of Amenta. Let (Formula presented.) be a family of compact sets in (Formula presented.) such that for any (Formula presented.), the intersection (Formula presented.) is either empty or contractible. It is shown that if (Formula presented.) is a family of sets such that for any finite (Formula presented.), the intersection (Formula presented.) is a union of at most r disjoint sets in (Formula presented.), then the Helly number of (Formula presented.) is at most r(d + 1).