Combinatorics I

Fall 2017

(642:582)

 

Course Info

Instructor: Swastik Kopparty (swastik.kopparty@rutgers.edu)

Class Time and Place: Tuesdays and Thursdays 1:40 to 3:00, in Hill 423

Office Hours: Mondays 3pm-4pm (Hill 432)

Prerequisites: Basic linear algebra, some discrete probability and mathematical maturity.

Reference books: Enumerative Combinatorics (Stanley), The Probabilistic Method (Alon & Spencer), Extremal Combinatorics (Jukna), A Course in Combinatorics (van Lint and Wilson).

 

 

Syllabus

 

We will study basic topics in Combinatorics, including enumeration, symmetry, polyhedral combinatorics, partial orders, set systems, Ramsey theory, discrepancy, additive combinatorics and quasirandomness. There will be emphasis on general techniques, including probabilistic methods, linear-algebra methods, analytic methods, topological methods and geometric methods.

 

There will be problem sets every 2-3 weeks. There are no exams.

 

Homework

·       Homework 1 (due October 3)

·       Homework 2 (due October 26)

·       Homework 3 (due November 16)

·       Homework 4 (due December 12)

 

Lecture Schedule

·       September 5: basic counting problems, the twelve-fold way, inclusion-exclusion

·       September 7: Stirling numbers, linear recurrences, generating functions

·       September 12: Bell numbers, Catalan numbers

·       September 14: more Catalan numbers

·       September 19: exponential generating functions, exponential formula

·       September 21: partitions of an integer

·       September 26: set systems, Hall’s theorem, posets, Dilworth’s theorem

·       September 28: the max-flow min-cut theorem

·       October 3: Class Cancelled (Makeup class to be scheduled)

·       October 5: Class Cancelled (Makeup class to be scheduled)

·       October 10: Sperner’s theorem, Littlewood-Offord, Erdos-Ko-Rado

·       October 12: Erdos-Ko-Rado 2, introduction to isoperimetric theorems

·       October 17: Kruskal-Katona

·       October 19: Kruskal Katona continued, Harper’s theorem

·       October 24: Katona’s intersection theorem, Ramsey theorem

·       October 26: Ramsey theorem continued

·       October 31: the probabilistic method

·       November 2: linearity of expectation

·       November 7: alterations

·       November 9: the second moment method

·       November 14: more application of probabilistic methods

·       November 16: the Lovasz Local Lemma

·       November 21: introduction to linear algebraic methods, Fisher’s inequality

·       November 23: NO CLASS (Thanksgiving)

·       November 28: two-distance sets, Ray Chaudhuri Wilson inequality

·       November 30: Frankl Wilson inequality, geometric applications

·       December 5: more applications

·       December 7: VC dimension, Sauer-Shelah Lemma epsilon-nets

·       December 12: discrepancy