A topological colorful Helly theorem (Q702702)

Let \(X\) denote a simplicial complex. If \(H_j(X)\) is the \(j\)th homology group of \(X\) with rational coefficients, then the reduced rational homology is denoted by \(\widetilde H_j(Y)\). Let \(d\) be a positive integer. If \(\widetilde H_i(Y) = 0\) for all induced subcomplexes \(Y\) of \(X\) and all \(i \geq d\), then \(X\) is called \(d\)-Leray. The main result of this paper is the following extension of the colorful Helly theorem to \(d\)-Leray complexes: if \(X\) is a \(d\)-Leray complex on a vertex set \(V\) and \(M\subset X\) is a matroidal complex on \(V\) with rank function \(\rho\), then there exists a simplex \(\tau \in X\) such that \(\rho(V-\tau) \leq d\).