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: Thursday 2:00-3:30 or by appointment


Course information: Postscript PDF

A sample final: 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

Homework 4: Postscript PDF
Hints for Homework 4: Postscript PDF


Schedule of paper presentations:

Wednesday April 24

Friday, April 26

Monday, April 29

Wednesday, May 1

Friday, May 3


Possible presentation papers:

D. Achlioptas and C. Moore, The Asymptotic Order of the k-SAT Threshold, Proc. Foundations of Computer Science (FOCS) 2002.

Y. Azar, A. Broder, A. Karlin, and E. Upfal, Balanced allocations, SIAM J. on Computing, 29, (2000), 180-200.

T. Bohman, A. Frieze and E. Lubetzky, A note on the random greedy triangle-packing algorithm , Journal of Combinatorics, 1 (2010), 477-488.

Svante Janson, The probability that a random multigraph is simple. Combin. Probab. Comput. 18 (2009), 205-225.

M. Krivelevich, Bounding Ramsey numbers through large deviation inequalities, Random Structures and Algorithms 7 (1995), 145-155.

A Nachmias, Y. Peres, The critical random graph, with martingales, Israel Journal of Math, 176 (2010) 29-43.

A Nachmias, Y. Peres, Component sizes of the random graph outside the scaling window , Latin American Journal of Probability and Mathematical Statistics (ALEA), 3, 133-142 (2007).

B. Reed and B. Sudakov, Asymptotically the list colouring constants are 1, J. Combinatorial Theory Ser. B 86 (2002), 27-37.

A. Rucinski and N. Wormald, Random graph processes with degree restrictions, Combinatorics, Probability and Computing 1 (1992) 169-180.

J. Spencer Asymptotic Packing via A Branching Process Random Structures and Algorithms, 7 (1995,) 167-172

J. Spencer and N. Wormald, Birth control for giants , Combinatorica 27 (2007), 587-628.