Leray numbers of tolerance complexes

Let $K$ be a simplicial complex on vertex set $V$. $K$ is called $d$-Leray if the homology groups of any induced subcomplex of $K$ are trivial in dimensions $d$ and higher. $K$ is called $d$-collapsible if it can be reduced to the void complex by sequentially removing a simplex of size at most $d$ that is contained in a unique maximal face. We define the $t$-tolerance complex of $K$, $\mathcal{T}_t(K)$, as the simplicial complex on vertex set $V$ whose simplices are formed as the union of a simplex in $K$ and a set of size at most $t$. We prove that for any $d$ and $t$ there exists a positive integer $h(t,d)$ such that, for every $d$-collapsible complex $K$, the $t$-tolerance complex $\mathcal{T}_t(K)$ is $h(t,d)$-Leray. The definition of the complex $\mathcal{T}_t(K)$ is motivated by results of Montejano and Oliveros on "tolerant" versions of Helly's theorem. As an application, we present some new tolerant versions of the colorful Helly theorem.