# Block design

## Definition

### Definition with symbols

A $(v,k,\lambda)$-block design is the following data:

• A set $S$ with $v$ elements, called the vertices
• A collection of $k$-element subsets of $S$, called blocks. The number of blocks is denoted as $b$

Satisfying the following conditions:

• Every point is contained in the same number of blocks (this is denoted as $r$)
• Given any two points, the number of blocks containing both of them is $\lambda$
Note that $b$ and $r$ are dependent on the other three parameters via some obvious relations PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

This is also called a BIBD or a Balanced Incomplete Block Design.

### Further definition

A generalization of block design is to $t$-block design. A $t$-block design has the property that for any $t$ elements, the number of blocks cotaining all $t$ of them is fixed and independent of the choice of elements. Such a thing is called, in more detail, a $t-(v,k,\lambda)$ block design.