Probabilistic Methods in Combinatorics - 0366.4913.01
Procedural Matters:
Prerequisite Courses:
Discrete Mathematics, Introduction to Probability.
Exercises will be given during the course and their solutions will
be graded.
Text books:
Most of the topics covered in the course appear in the
books listed below (especially the first one). Other topics appear in
recent papers, many of which can be found in the journal
Random Structures and Algorithms.
N. Alon and J. H. Spencer,
The Probabilistic Method,
Wiley, 1992. (Second Edition, 2000, Third Edition 2008).
B. Bollobas,
Random Graphs,
Academic Press, 1985. (Second Edition, Cambridge University Press, 2001.)
S. Janson, T. Luczak and A. Rucinski,
Random Graphs,
Wiley, 2000.
M. Molloy and B. Reed,
Graph Colouring and the Probabilistic Method,
Springer,
2002.
Course syllabus:
Probabilistic methods in Combinatorics and their
applications in theoretical Computer Science. The topics include
linearity of expectation, the second moment method, the local
lemma, correlation inequalities, martingales, large deviation
inequalities, geometry, derandomization.
Course Outline (to be updated during the term):
-
: October 22
The basic method, Ramsey numbers, Dominating sets in
graphs, Hypergraph 2-coloring, Sum-free subsets,
Set pairs theorem.
-
: October 29
Linearity of expectation, Hamiltonian paths in
tournaments and the conjecture of Szele, Minc Conjecture.
-
: November 5
The second moment method, Turan's proof of
the Hardy Ramanujan theorem, distinct sums, random graphs and
threshold functions, cliques in random graphs.
-
: November 12
More on cliques in random graphs.
Alterations: Graphs with high girth and high chromatic number.
Bounding of large deviations and
consistent arcs in tournaments.
-
: November 19
Dependent Random Choice and its applications (guest lecture by W. Samotij)
-
: November 26
The local lemma: the general lemma and its symmetric version,
Straus' problem, directed cycles.
-
: December 3
More on the local lemma, solution of (some) exercises.
-
: December 10
Correlation inequalities:
the four functions theorem and its
applications, the FKG Inequality, Kleitman's Theorem, correlation
between properties of random graphs.
-
: December 17
Martingales: the edge exposure and the vertex exposure
martingales, Azuma's Inequality.
-
: December 24
More martingales, chromatic number of sparse random graphs.
-
: December 31
Poisson approximation: The Janson Inequalities and their
application for constructing Ramsey type graphs.
-
: January 7
No class
-
: January 14
The chromatic number of G(n,1/2).
Geometry: the VC dimension of a range space and its applications.
-
: January 21
Dominating sets in k-majority tournaments, summary.
Exercises