Block design

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

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.