Occupancy fraction, fractional colouring, and triangle fraction

Given $$ , there exists $$ such that, if $$ , then for any graph $$ on $$ vertices of maximum degree $$ in which the neighbourhood of every vertex in $$ spans at most $$ edges, (i)an independent set of $$ drawn uniformly at random has at least $$ vertices in expectation, and(ii)the fractional chromatic number of $$ is at most $$ . These bounds cannot in general be improved by more than a factor 2 asymptotically. One may view these as stronger versions of results of Ajtai, Komlós and Szemerédi and Shearer. The proofs use a tight analysis of the hard-core model.