Combinatorics I

Fall 2012

(642:582)

 

Course Info

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

Class Time and Place: Mondays and Wednesdays 1:40 to 3:00, in Hill 423

Office Hours: Wednesday 3:30-4:30 (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 September 24)

·         Homework 2 (due October 22)

·         Homework 3 (due November 19)

·         Homework 4 (due December 5)

 

 

Lecture Schedule

·         September 5: course overview, basic counting problems

·         September 10: linear recurrences, the twelvefold way, Stirling numbers

·         September 12: Catalan numbers, generating function nuggets

·         September 17: theory of exponential generating functions

·         September 19: permutations

·         September 24: partitions of an integer (notes)

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

·         October 1: Sperner’s theorem, LYM inequality, Erdos-Ko-Rado 1

·         October 3: Erdos-Ko-Rado 2, Littlewood-Offord

·         October 8: compressions, Kruskal-Katona

·         October 10: vertex and edge isoperimetry in {0,1}ⁿ (notes)

·         October 15: Ramsey theory

·         October 17: Class cancelled (makeup class at the end of the semester)

·         October 22: the probabilistic method, linearity of expectation, alterations

·         October 24: Turan’s theorem, Turan number for C4, the projective plane

·         October 29: NO CLASS (Hurricane)

·         October 31: NO CLASS (Hurricane’s aftermath)

·         November 5: linear algebraic methods, few-distance sets, restricted intersections

·         November 7: modular intersections, geometric applications

·         November 12: Class cancelled (makeup class at the end of the semester)

·         November 14: Class cancelled (makeup class at the end of the semester)

·         November 19: projections of set systems, Sauer-Shelah lemma, VC dimension

·         November 21: NO CLASS (Friday Schedule)

·         November 26: the second moment

·         November 28: entropy, Shearer’s lemma, Bregman’s theorem

·         December 3: the Lovasz local lemma

·         December 5: discrepancy

·         December 10: Baranyai’s theorem (notes)

·         December 12: sunflowers and miscellaneous gems

·         December 13 (1pm – 4pm, Hill 423):  Makeup marathon

§  introduction to additive combinatorics: van der Waerden’s theorem, Roth’s theorem, Schur’s theorem, Sidon sets, Cauchy-Davenport theorem

§  topological methods: Borsuk-Ulam theorem, Kneser’s conjecture