Occupancy fraction, fractional colouring, and triangle fraction.

Abstract

Given ε>0, there exists f0 such that, if f0≤f≤Δ2+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 Δ2∕f edges, (i)an independent set of G drawn uniformly at random has at least (1∕2−ε)(n∕Δ)logf vertices in expectation, and(ii)the fractional chromatic number of G is at most (2+ε)Δ∕logf. 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.