Coloring Graphs with Dense Neighborhoods

It is shown that any graph with maximum degree in which the average degree of the induced subgraph on the set of all neighbors of each vertex exceeds 6k26k2+1+k+6 is either (-k)-colorable or contains a clique on more than -2k vertices. In the k=1 case we improve the bound on the average degree to 23+4 and the bound on the clique number to -1. As corollaries, we show that every graph satisfies max{,-1,4} and every graph satisfies max{,-1,remvoe15+48n+734}.