On Turáns theorem for sparse graphs

a?nl(t+1) where a denotes the maximum size of an independent set in G. We improve this bound for graphs containing no large cliques. 0. Notation n=n(G)=number of vertices of the graph G e=e(G)=number of edges of G h=h(G)=number of triangles in G deg (P)=valency (degree) of the vertex P deg, (P) = triangle-valency of P=number of triangles in G adjacent to P t=t(G)=n f deg (P) = 2eln = average valency in G (we will tacitly assume t' 1) T= T(G) =maximum valency in G a=a(G)=maximum size of independent set of vertices (independence or stability number) Kp =shorthana'-for p-clique log x=max {1, In x} to , cl , c2,. .. are absolute constants when speaking of union, difference or partition of graphs, we work with the vertex-sets 1. Introduction Let G be a graph of n vertices and a edges with average valency t=2eln. It is an easy consequence of the celebrated Turán's theorem [6] (and can easily be proved directly) that G contains an independent set of size nl(t+1), i .e. a-nl(t+1) .