We analyse the asymptotic behaviour of random instances of the maximum set packing (MSP) optimization problem, also known as maximum matching or maximum strong independent set on hypergraphs. We give an analytic prediction of the MSPs size using the 1RSB cavity method from statistical mechanics of disordered systems. We also propose a heuristic algorithm, a generalization of the celebrated Karp-Sipser one, which allows us to rigorously prove that the replica symmetric cavity method prediction is exact for certain problem ensembles and breaks down when a core survives the leaf removal process. The e-phenomena threshold discovered by Karp and Sipser, marking the onset of core emergence and of replica symmetry breaking, is elegantly generalized to Cs=e/(d-1) for one of the ensembles considered, where d is the size of the sets.