Interpolating between bounds on the independence number

For a non-negative integer T , we prove that the independence number of a graph G = ( V , E ) in which every vertex belongs to at most T triangles is at least ∑ u ∈ V f ( d ( u ) , T ) where d ( u ) denotes the degree of a vertex u ∈ V , f ( d , T ) = 1 d + 1 for T ≥ d 2 and f ( d , T ) = ( 1 + ( d 2 − d − 2 T ) f ( d − 1 , T ) ) / ( d 2 + 1 − 2 T ) for T < d 2 . This is a common generalization of the lower bounds for the independence number due to Caro, Wei, and Shearer. We discuss further possible strengthenings of our result and pose a corresponding conjecture.