Occupancy fraction, fractional colouring, and triangle fraction
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Given ε>0, there exists f_0 such that, if f_0 < f <Δ^2+1, then for any graph G on n vertices of maximum degree Δ in which the neighbourhood of every vertex in G spans at most Δ^2/f edges, (i) an independent set of G drawn uniformly at random has at least (1/2-ε)(n/Δ) f vertices in expectation, and (ii) the fractional chromatic number of G is at most (2+ε)Δ/ f. 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 (1981) and Shearer (1983). The proofs use a tight analysis of the hard-core model.
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