Time and Place:
Monday and Wednesday 3:00-4.50 P.M., ROYCE 156.
First class - October 1
Benny Sudakov, MS 7911, email@example.com
The Probabilistic Method is a powerful tool in tackling
many problems in discrete mathematics. It belongs to those areas of mathematics
which have experienced a most impressive growth in the past few decades.
Roughly speaking, its basic idea can be described as follows. In order
to prove existence of a combinatorial structure with certain properties,
we construct an appropriate probability space, and show that a randomly
chosen element of this space has the desired property, with positive probability.
This course provides a gentle introduction to the Probabilistic
Method, with emphasis on methodology. We will try to illustrate the main
ideas by showing the application of probabilistic reasoning to various
combinatorial problems. The topics covered in the class will include (but
are not limited to):
Linearity of expectation, the second
moment method, the local lemma, correlation inequalities, martingales,
large deviation inequalities, Janson and Talagrand inequalities and pseudo-randomness.
Most of the topics covered in the course appear in the
books listed below (especially the first one). Other topics appear in recent
The Probabilistic Method, by N. Alon and J. H.
Spencer, 3rd Edition, Wiley, 2008.
Random Graphs, by B. Bollobas, 2nd Edition, Cambridge
University Press, 2001.
Random Graphs, by S. Janson, T. Luczak and A. Rucinski,
Graph Coloring and the Probabilistic Method, by
M. Molloy and B. Reed, Springer, 2002.
During this term there will be 3 homework in this course. Students are required to submit them
to get a final grade.
Course Outline (to be updated during
The basic method: Ramsey numbers,
Set-pairs with restricted intersections, Hypergraph 2-coloring,
Dominating sets in graphs. Application of dominating sets in graphs to covering codes.
Linearity of expectation: Max Cut in graphs.
Maximum and minimum
number of Hamilton paths in tournament and
and the conjecture of Szele.
Permanents of matrices - Minc Conjecture.
Existence of graphs with large girth and large chromatic number.
Recoloring and new bounds on the minimum number of edges in
non-2-colorable n-uniform hypergraphs.
The second moment method: Turan's proof of the Hardy
Numbers with all distinct sums, random graphs and threshold functions, cliques in
Bounding large deviations (Chernoff bounds) and consistent arcs in
Lovasz Local Lemma: application to 2-colorability of n-uniform hypergraphs
Proof of Lovasz Local Lemma, Applications of Lovasz Local Lemma to:
cycles in directed graphs, acyclic edge coloring, coloring of integers which makes all shifts
of given set multicolored.
Correlation inequalities: 4 function theorem and proof of
FKG inequality, application to monotone properties in random graphs.
Large deviation inequalities
for martingales. Chromatic number of random graphs. Isoperimetric inequality for Hamming cube.
Probabilistic gems: Ramsey and Turan numbers of sparse graphs, Independence number of triangle-free
graph, crossing numbers, incidence and additive number