# Block design

From Groupprops

## Definition

### Definition with symbols

A -block design is the following data:

- A set with elements, called the
*vertices* - A collection of -element subsets of , called blocks. The number of blocks is denoted as

Satisfying the following conditions:

- Every point is contained in the same number of blocks (this is denoted as )
- Given any two points, the number of blocks containing both of them is

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

### Further definition

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