- Lecturer: Alexander Scott

General Prerequisites:

B8.5 Graph Theory is helpful, but not required.

Course Term: Michaelmas

Course Lecture Information: 16 lectures

Course Weight: 1

Course Level: M

Assessment Type: Written Examination

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. Isoperimetry in the cube.

Intersecting families. Erdos-Ko-Rado Theorem. Cross-intersecting families. t-intersecting families. Fisher's Inequality. Frankl-Wilson Theorem. Application to Borsuk's Conjecture.

VC-dimension. Sauer-Shelah Theorem.

Combinatorial Nullstellensatz.

Shadows. Kruskal-Katona Theorem. Isoperimetry in the cube.

Intersecting families. Erdos-Ko-Rado Theorem. Cross-intersecting families. t-intersecting families. Fisher's Inequality. Frankl-Wilson Theorem. Application to Borsuk's Conjecture.

VC-dimension. Sauer-Shelah Theorem.

Combinatorial Nullstellensatz.