Institute for Advanced Studies in Basic Sciences (IASBS)
Probabilistic graph theory
MATH M 201

Fall 2015
7. {Alon—Spencer #1} Prove that if there is a real 1:), D E p E 1, so that
(:)p121+(:){1 was .1: 1,
then the Ramsey number T(l:,t} satisﬁes :r'(:l:,t) 2} 11. Conclude that
114,1) 2 11(13f2/(10g113/2).
Solution (WeiTian Li): Coloring the edges of Kn rando
Institute for Advanced Studies in Basic Sciences (IASBS)
Probabilistic graph theory
MATH M 201

Fall 2015
Problem Set 1: Solutions
MATH 778P, Fall 2008, Cooper
Expiration: Monday September 15
You are awarded up to 12 points for each problem, 5 points for submitting
A
solutions in L TEX, and 5 points per solution that is used for the answer key.
1.
For a set o
Institute for Advanced Studies in Basic Sciences (IASBS)
Probabilistic graph theory
MATH M 201

Fall 2015
CHAPTER 2 OF THE PROBABILISTIC METHOD*
CESAR CUENCA
Problem 1: Suppose n 2 and let H = (V, E) be an nuniform hypergraph with E = 4n1 edges. Show
that there is a coloring of V by four colors so that no edge is monochromatic.
Solution: Color each vertex
Institute for Advanced Studies in Basic Sciences (IASBS)
Probabilistic graph theory
MATH M 201

Fall 2015
Theorem 3.5. Let G be a graph with. n nodes and m 2 71/2 edges. Then
2
a(G) 3 4m.
Proof. We set p = act/(2m). By assumption 0 g p g 1. Let V = {1:1, 'Ug: . . . , on}
denote the vertex set of G. We now choose a random subset S of V where
each vertex is cho