GIVEN a set X and a natural number r denote by X(r) the set of relement subsets of X . An r-graph or hypergraph G is a pair (V, T), where V is a finite set and T c V(r) . We call v E V a vertex of G and z c-T an r-tuple or an edge of G . Thus a 1-graph is a set V and a subset T of V. As the structure of 1-graphs is trivial, throughout the note we suppose r > 2 . A 2-graph is a graph in the sense of (5) . The graph Er (n, k) clearly does not contain k+1 independent r-tuples and it is maximal with this property if n > ( k + 1)r . Let us define another maximal r-graph with at most k independent r-tuples, Fr (n, k) = (Vi, T1 ) . Let IV, = n > k+r, let W1 and R be disjoint subsets of V1 , I W1I = k-1, IRI = r, and let v E V1-W1-R . Then the set of r-tuples of F r (n, k) is T 1 = {,C EVm :ti n W 1~0} u {ze V (r) :vezand-rnR 0 } v {R} .

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For r-partite and for r-uniform hypergraphs bounds are given for the minimum degree which ensures d independent edges.

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