In mathematics, the Turán number T(n,k,r) for r-uniform hypergraph

Hypergraph

In mathematics, a hypergraph is a generalization of a graph, where an edge can connect any number of vertices. Formally, a hypergraph H is a pair H = where X is a set of elements, called nodes or vertices, and E is a set of non-empty subsets of X called hyperedges or links...

s 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. ́

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 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.