Dense H-free graphs are almost (\chi(H)-1)-partite
Combinatorics
2011-02-17 v3
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.
Cite
@article{arxiv.0907.3815,
title = {Dense H-free graphs are almost (\chi(H)-1)-partite},
author = {Peter Allen},
journal= {arXiv preprint arXiv:0907.3815},
year = {2011}
}
Comments
18 pages, no figures. Published in Electronic Journal of Combinatorics without appendix (added after publication) detailing what can be done for general monotone properties