C8.3 Combinatorics - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Jason Long
General Prerequisites: 

Part B Graph Theory is helpful, but not required.

Course Term: 
Michaelmas
Course Lecture Information: 

16 lectures

Course Weight: 
1.00 unit(s)
Course Level: 
M

Assessment type:

Course Overview: 

An important branch of discrete mathematics concerns properties of collections of subsets of a finite set. There are many beautiful and fundamental results, and there are still many basic open questions. The aim of the course is to introduce this very active area of mathematics, with many connections to other fields.

Learning Outcomes: 

The student will have developed an appreciation of the combinatorics of finite sets.

Course Synopsis: 

Chains and antichains. Sperner's Lemma. LYM inequality. Dilworth's Theorem.

Shadows. Kruskal-Katona Theorem.

Intersecting families. Erdos-Ko-Rado Theorem. Cross-intersecting families.

VC-dimension. Sauer-Shelah Theorem.

t-intersecting families. Fisher's Inequality. Frankl-Wilson Theorem. Application to Borsuk's Conjecture.

Combinatorial Nullstellensatz.

Reading List: 
  1. Bela Bollobás, Combinatorics, CUP, 1986.
  2. Stasys Jukna, Extremal Combinatorics, Springer, 2007