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, and the problem for general r was introduced in . The paper gives a survey of Turán numbers.
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 (n ≥ k ≥ r) 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.
The complements of the lines of the Fano plane form a Turán (7,5,4)-system. T(7,5,4) = 7.
It can be shown that
T(n,k,r)\geq\binom{n}{r}{\binom{k}{r}}-1.
An (n,r,k,r)-lotto design is an (n, k, r)-Turán system. Thus, T(n,k, r) = L(n,r,k,r).