 
Summary: University of Washington Math 523A Lecture 5
Lecturer: Eyal Lubetzky
Monday, April 13, 2009
1 Review: Doob's martingales on G(n, p)
Let G = (V, E) be a graph on n vertices and f a function on such graphs. The vertex set of
G is V = {v1, . . . , vn}, the edge set E is a subset of {e1, . . . , em}, where m = n
2
. Suppose
that G G(n, p), i.e. the edges of G are IID Bernoulli(p). Last time we saw two special cases
of Doob's martingale process applied to G(n, p), namely the edge exposure martingale
Xt = E[f(G)  1{e1G}, . . . , 1{etG}],
and the vertex exposure martingale
Yt = E[f(G)  G{v1,...,vt+1}].
Here G{v1,...,vt+1} denotes the induced subgraph on {v1, . . . , vt+1}. Note that for the vertex
exposure martingale Yt, the vertex vt+1 is revealed at time t, along with the t edges (or
nonedges) connecting vt+1 to each vertex in {v1, . . . , vt}.
Last time we saw that for f = (the chromatic number of G), the vertex exposure
martingale Yt satisfies Yt  Yt1 1, which allowed us to apply the HoeffdingAzuma
inequality to prove that n, p, if G G(n, p), then P ((G)  E(G) > a
