Dense H-free graphs are almost (\chi(H)-1)-partite

TL;DR

This paper proves that dense graphs avoiding a specific (r-1)-partite subgraph are close to being r-partite, improving previous results by avoiding the use of the Szemerédi Regularity Lemma.

Contribution

It establishes a stronger, more direct proof that dense graphs avoiding certain subgraphs are nearly r-partite, without relying on the Regularity Lemma.

Findings

Graphs with high minimum degree either contain the subgraph H or are close to r-partite after removing few edges.

The result applies to any (r-1)-partite graph H with a smallest part of size t.

Provides a bound on edges to delete to achieve r-partiteness, depending on t.

Abstract

By using the Szemer\'edi Regularity Lemma, Alon and Sudakov recently extended the classical Andr\'asfai-Erd\~os-S\'os theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r-1)-partite graph H whose smallest part has t vertices, and any fixed c>0, there exists a constant C such that whenever G is an n-vertex graph with minimum degree at least ((3r-4)/(3r-1)+c)n, either G contains H, or we can delete at most Cn^(2-1/t) edges from G to yield an r-partite graph.