  The first author together with Jenssen, Perkins and Roberts (2017) recently
showed how local properties of the hard-core model on triangle-free graphs
guarantee the existence of large independent sets, of size matching the
best-known asymptotics due to Shearer (1983). The present work strengthens this
in two ways: first, by guaranteeing stronger graph structure in terms of
colourings through applications of the Lov\'asz local lemma; and second, by
extending beyond triangle-free graphs in terms of local sparsity, treating for
example graphs of bounded local edge density, of bounded local Hall ratio, and
of bounded clique number. This generalises and improves upon much other earlier
work, including that of Shearer (1995), Alon (1996) and Alon, Krivelevich and
Sudakov (1999), and more recent results of Molloy (2019), Bernshteyn (2019) and
Achlioptas, Iliopoulos and Sinclair (2019). Our results derive from a common
framework built around the hard-core model. It pivots on a property we call
local occupancy, giving a clean separation between the methods for deriving
graph structure with probabilistic information and verifying the requisite
probabilistic information itself.
