21-737 Probabilistic Combinatorics

This course covers the probabilistic method for combinatorics in detail and introduces randomized algorithms and the theory of random graphs.

Methods covered include the second moment method, the R\"odl nibble, the Lov\'asz local lemma, correlation inequalities, martingale's and tight concentration, Janson's inequality, branching processes, coupling and the differential equations method for discrete random processes. Objects studied include the configuration model for random regular graphs, Markov chains, the phase transition in the Erd\H{o}s-R\'enyi random graph, and the Barab\'asi-Albert preferential attachment model.

Course Instructor:

Tom Bohman
Wean Hall 6105
tbohman@math.cmu.edu
Office Hours: Wednesday 3:00-4:0 or by appointment


Course information: Postscript PDF

Homework 1: Postscript PDF
Hints for Homework 1: Postscript PDF

Homework 2: Postscript PDF
Hints for Homework 2: Postscript PDF

Homework 3: Postscript PDF
Hints for Homework 3: Postscript PDF


Tentative schedule:

Week 1: Alon and Spencer, Probabilistic Lenses `Triangle-free graphs have large independence numbers' and `Crossing numbers, incidences, sums and products'.

Week 2: Second moment method. See Alon and Spencer Sections 4.5 Clique Number and 4.6 Distinct Sums.

Week 3: Second moment method. Satisfiability threshold for random 2-SAT. See Chvatal and Reed: Mick gets some (the odds are on his side), FOCS, 1992, pp. 620--627.

Week 4: Lovasz Local Lemma (and linear arboricity). Alon and Spencer: Sections 5.1, 5.2, 5.5

Week 5: The Moser-Tardos Local Lemma Algorithm. See Moser and Trados: A constructive proof of the general Lovász Local Lemma, Journal of the ACM 57 (2010) (2) Art. 11, 15pp. Also see some notes written by Joel Spencer.

Week 6: Janson's Inequality. See Alon and Spencer Section 8.1-8.3. Diameter of a random graph.

Week 7: Martingales and tight concentration. See Alon and Spencer 7.1-7.4.

Week 8: An Introduction to Dynamic Concentration

Week 9: Branching processes and the giant component. See Janson, Luczak and Rucinski Section 5.2

Week 10: Applictaions of Dynamic concentration. Giant Component