Degree Powers in Graphs: The Erdős–Stone Theorem

Abstract

Let 1 ≤p≤r+ 1, withr≥ 2 an integer, and letGbe a graph of ordern. Letd(v) denote the degree of a vertexv∈V(G). We show that ifthenGhas more than(r+ 1)-cliques sharing a common edge. From this we deduce that ifthenGcontains more thancliques of orderr+ 1.In turn, this statement is used to strengthen the Erdős–Stone theorem by using ∑v∈V(G)dp(v) instead of the number of edges.