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.
The paper (Sidorenko 1995) gives a survey of Turán numbers.
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.
[1] It can be shown that Equality holds if and only if there exists a Steiner system S(n - k, n - r, n).