Hypergraphs
Professor Andrew Thomason
We defined ex(n, F ) and (F ) for hypergraphs. Let Krl be the complete l-uniform hypergraph
on r vertices. No value of (Krl ) is known for r > l 3.
Tur
an conjectured that (K43 ) = 5/9. Erdos offered $5000 in his memo

Extremal graph theory
Professor Andrew Thomason
2. Chapter II. Stability
An extremal problem is stable if every near-optimal (that is, having almost ex() edges)
example is very close to some specific extremal graph. [Intuition: we can think of that as say

EXTREMAL GRAPH THEORY
PROFESSOR ANDREW THOMASON
dis regularity lemma
3. Chapter 3. Szemere
Its cool because it tells us something about the structure of every graph, and has very useful
applications to e.g. arithmetic number theory.
A graph with the (larg

Extremal graph theory
Professor Andrew Thomason
4. Chapter 4: Containers
We consider r-uniform hypergraphs G of order N and average degree d (typically we need d
large to get useful results).
So each edge is a subset of the vertex set of size r. The degre

EXTREMAL GRAPH THEORY
PROFESSOR ANDREW THOMASON
s-Stone theorem
1. Chapter 1. The Erdo
1.1. Tur
ans theorem. We begin with a basic result from extremal graph theory that will be
the basis for much generalization.
Theorem 1.1 (Tur
ans theorem, also due to