  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.
