# Every graph with $n\ge3$ vertices and more than $\frac{n^2}{4}$ edges contains a triangle

Prove:

(1) Every graph with $n\ge3$ vertices and more than $\frac{n^2}{4}$ edges contains a triangle.

(2) For every even $n$ there exists a graph with $n$ vertices and $\frac{n^2}{4}$ edges that doesn't contain a triangle.

I know this is a well-known theorem by Turan, but I can't understand the proof there.

• If you have a particular proof that you do not understand, it may be useful for you to reproduce the proof and note what parts you are having difficulty understanding. – Aaron Jan 31 '18 at 11:55
• This smells like the pigeonhole principle from $200$ miles away. – Arnaud Mortier Jan 31 '18 at 13:25

## 1 Answer

Well (2) is easy: Say the vertices are the union of two disjoint sets $V_1$ and $V_2$, each with $n/2$ elements, and add edges $(v,w)$ for every $v\in V_1$ and $w\in V_2$.

A hint for (1): If $S$ is a set containing exactly three vertices let $E_S$ be the set of all edges from one vertex in $S$ to another vertex in $S$. You need to show that some $E_S$ contains three edges.