An extension of the Erdős-Stone theorem
The following extension of the Erdős-Stone theorem is presented: There is an absolute constant \(\alpha> 0\) such that, for all \(r\geq 1\), \(0< \beta< 1\) and \(0< \varepsilon\leq 1/r\), every graph \(G\) of sufficiently large order \(n\) with at least \((1- {1\over r}+ \varepsilon)(\begin{smallmatrix} n\\ 2\end{smallmatrix})\) edges contains a complete \((r+ 1)\)-partite graph \(K_{r+ 1}(s,m,\dots, m,t)\), where \(m= m(n)= \lfloor \alpha(1- \beta)(\log n)/\log r\rfloor\), \(s= s(n)= \lfloor \alpha(1- \beta)(\log n)r\log (1/\varepsilon)\rfloor\), and \(t= t(n)= \lfloor a\varepsilon^{1+ \beta/2} n^ \beta\rfloor\).
scientific article; zbMATH DE number 700950