# Turán number

In mathematics, the Turán number T(n,k,r) for r-uniform hypergraphs of order n is the smallest number of r-edges such that every induced subgraph on k vertices contains an edge. This number was determined for r = 2 by Turán (1941), and the problem for general r was introduced in Turán (1961). The paper (Sidorenko 1995) gives a survey of Turán numbers.

## Definitions

Fix a set X of n vertices. For given r, an r-edge or block is a set of r vertices. A set of blocks is called a Turán (n,k,r) system (nkr) if every k-element subset of X contains a block. The Turán number T(n,k,r) is the minimum size of such a system.

## Example

The complements of the lines of the Fano plane form a Turán (7,5,4)-system. T(7,5,4) = 7.

## Relations to other combinatorial designs

It can be shown that

$T(n,k,r)\geq {\binom {n}{r}}{\binom {k}{r}}^{-1}.$ Equality holds if and only if there exists a Steiner system S(n - k, n - r, n).

An (n,r,k,r)-lotto design is an (n, k, r)-Turán system. Thus, T(n,k, r) = L(n,r,k,r).