q-Overlaps in the Random Exact Cover Problem

We prove upper and lower bounds for the threshold of the q-overlap-k-Exact cover problem. These results are motivated by the one-step replica symmetry breaking approach of Statistical Physics, and the hope of using an approach based on that of Mezard et al. (2005) to rigorously prove that for some values of the order parameter the overlap distribution of k-Exact Cover has discontinuous support.