Independent sets in graphs with triangles

In this note we give a fast algorithm which computes an independent set of size at least Omega((n/Delta) In Delta) for a graph G on n vertices with maximum degree Delta, if G contains only a little less than the maximum possible number of triangles (say n Delta(2-epsilon) many for a positive constant epsilon). Our algorithm uses derandomization and is based on earlier results concerning the independence number of triangle-free graphs due to Ajtai et al. (1980) and Shearer (1983).