Large Multipartite Subgraphs in H-free Graphs

Abstract

In this work, we discuss a strengthening of a result of Füredi that every n-vertex $K_{r+1}$-free graph can be made r-partite by removing at most $T(n,r) - e(G)$ edges, where $T(n,r)=\frac{r-1}{2r}n^2$ denotes the number of edges of the n-vertex r-partite Turán graph. As a corollary, we answer a problem of Sudakov and prove that every $K_6$-free graph can be made bipartite by removing at most $4n^2/25$ edges. The main tool we use is the flag algebra method applied to locally definied vertex-partitions.