Occupancy fraction, fractional colouring, and triangle fraction

Given epsilon > 0, there exists f 0 such that, if f 0 <= f <= Delta 2 + 1, then for any graph G on n vertices of maximum degree Delta in which the neighbourhood of every vertex in G spans at most Delta 2 / f edges, an independent set of G drawn uniformly at random has at least ( 1 / 2 - epsilon ) ( n / Delta ) log f vertices in expectation, and the fractional chromatic number of G is at most ( 2 + epsilon ) Delta / log f. (i)(ii) 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, Komlos and Szemeredi and Shearer. The proofs use a tight analysis of the hard-core model.