Topic Course on
Probabilistic Methods
(Week 1)
Linearity of Expectation (1)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook The
probabilistic Method (third edition
Topic Course on
Probabilistic Methods
(Week 6)
Correlation Inequalities
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook The
probabilistic Method (third edition) by
Topic Course on
Probabilistic Methods
(Week 5)
Lovsz Local Lemma
a
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook The
probabilistic Method (third edition) by Noga
Topic Course on
Probabilistic Methods
(Week 7)
Large deviation inequalities (I)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook The
probabilistic Method (third edi
Topic Course on
Probabilistic Methods
(Week 8)
Large deviation inequalities (II)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook The
probabilistic Method (third ed
Topic Course on
Probabilistic Methods
(Week 10)
Poisson Paradigm
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook The
probabilistic Method (third edition) by Noga A
Topic Course on
Probabilistic Methods
(Week 9)
Large deviation inequalities (III)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook The
probabilistic Method (third e
Topic Course on
Probabilistic Methods
(Week 4)
Second Moment Method
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook The
probabilistic Method (third edition) by Nog
Topic Course on
Probabilistic Methods
(Week 3)
Alterations
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook The
probabilistic Method (third edition) by Noga Alon an
Math778P Homework 2 Solution
Choose any 5 problems to solve.
1. Let Sn = n Xi where X1 , . . . , Xn are n independent uniform cfw_1, 1
i=1
random variables. Prove that
E(|Sn |) = n21n
n1
n1
2
.
Proof by Danny Rorabaugh: Let Sn = n Xi where X1 , . . . , Xn
Math778P Homework 1 Solution
1. Prove Ramsey number R(4, t) (t3/2 /(ln t)3/2 ).
Proof by Heather Smith:
Claim: For a probability p, if
n.
n
4
4
p(2) +
n
t
t
(1 p)(2) < 1 then R(4, t) >
Randomly two color Kn where edges are colored red with probability
p a
Math778P Homework 3
due Oct. 5, Friday lecture
Choose any 5 problems to solve.
1. Use alteration method to prove the Ramsey number
2
t
ln t
R(4, t) c
for some constant c > 0.
Proof by Edward Boehnlein: By theorem 3.3.2, we have for any
p [0, 1] and n N,
n
Math778P Homework 4 Solution
1. Let G = (V, E ) be a simple graph and suppose each v V is associated
with a set S (v ) of colors of size at least 10d, where d 1. Suppose, in
addition, that for each v V and c S (v ) there at most d neighbors
u of v such th
Topic Course on
Probabilistic Methods
(Week 2)
Linearity of Expectation (2)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook The
probabilistic Method (third edition
Topic Course on
Probabilistic Methods
(Week 11)
Random Graphs (I)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook The
probabilistic Method (third edition) by Noga