 
Summary: Stability type results for hereditary properties
Noga Alon
Uri Stav
November 12, 2007
Abstract
The classical Stability Theorem of Erdos and Simonovits can be stated as follows. For a
monotone graph property P, let t 2 be such that t + 1 = min{(H) : H / P}. Then any
graph G
P on n vertices, which was obtained by removing at most (1
t + o(1)) n
2 edges from
the complete graph G = Kn, has edit distance o(n2
) to Tn(t), the TurŽan graph on n vertices
with t parts.
In this paper we extend the above notion of stability to hereditary graph properties. It turns
out that to do so the complete graph Kn has to be replaced by a random graph. For a hereditary
graph property P, consider modifying the edges of a random graph G = G(n, 1/2) to obtain a
graph G
that satisfies P in (essentially) the most economical way. We obtain necessary and
sufficient conditions on P which guarantee that G
