**MSC:**- 05D05 Extremal set theory
- 06A07 Combinatorics of partially ordered sets

is called {\em $t$-intersecting in $[s]$} if

\[

|X_1\cap X_2\cap [s]|\ge t \text{ for all } X_1, X_2\in\mathcal{F} .

\]

Let $\omega:2^{[n]}\rightarrow \mathbb{R}_{+}$ be a given weight function and

\[

M_s(n,t;\omega):=\max\{\omega(\mathcal{F}) : \mathcal{F} \text{ is $t$-intersecting in $[s]$}\} .

\]

For several weight functions, the numbers $M_n(n,t;\omega)$ can be determined using three important methods

of Ahlswede and Khachatrian: Generating Sets \cite{ahlk97}, Comparison Lemma \cite{ahlk98}, and

Pushing--Pulling \cite{ahlk97a}. We survey these methods.

Also, sufficient conditions on $\o$ for the equality

\[

M_s(n,t;\omega)=M_n(n,t;\omega)

\]

are presented which simplify the method of Generating Sets.

In addition, analogous conditions are given for the case that $|\cap_{X\in\mathcal{F}}X|

Applications of these methods include new intersection theorems for chain-- and star products.