Hard-Core Distributions on Matchings

In the preceding chapter, we analyzed two distributions on stable sets. In the first, each stable set was equally likely. In the second, each maximum stable set was equally likely. Our analyses allowed us to find large stable sets and fractional colourings using few colours. In both cases, the analysis involved showing that certain properties held in the neighbourhood of a vertex, regardless of what the stable set looked like further away from the vertex. This makes these two distributions attractive candidates for use in conjunction with the Local Lemma, for their local analysis leads to global results.