Local Density in Graphs with Forbidden Subgraphs

Abstract

A celebrated theorem of Turán asserts that every graph on n vertices with more than $ and every αn vertices of G span more than\[ \frac{r\,{-}\,1}{2r}(2\alpha -1)n^2\vspace*{7pt} \]edges, then G contains a copy of $ edges, is a Turán graph. We also obtain the local density version of the Erdős–Stone theorem.