A generalizect graph consists of a set of n vertices and a collection of k-tuples. of these vertices (cf. TURAN [1]). In what follows we shall refer to such a configuration as an edge-grapk if k = 2 and, usually, simply as a graph if k > 2. A complete m-graph has mvertices and [k)k-tuples. We say that a graph G is m-saturated if it contains no complete m-graph but loses this property when any new k-tuple is added.Tu~AN [2] proved the following theorem on edge-graphs in 1941: Let n = = g(m-1)+r, where g, m, and r are integers such that g=>l, m=>3, O<=r<=m -1, and n ~m. Then an m-satfirated edge-graph of n vertices can have at most Em= 2(in 1) (ha -r2) § edges. The dual problem was recently solved by ERI~6S, HAJNAL, and MOON [3] who showed that such an edge-graph must have at least e,,,=(m-2)(n-m § 1) edges. These two results can be combined as follows: If G is an m-saturated edge--graph of n Vertices and e edges, then em ~ e -<_ E,,,.The extremal edge-graphs for which e = % or e = E,~, are also characterized in these papers.Corresponding problems can be stated for generalized graphs. Let G be a (k +/)-saturated graph with n vertices and t k-tuples, where it is understood that k + l~ n. Let the maximum and minimum values of t over the class of such graphs G be denoted by Tk, z and t~.t, respectively.

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ABSTRACT:We present a few results and a larger number of questions concerning partitions of graphs or hypergraphs, where the objective is to maximize or minimize several quantities simultaneously. We consider a variety of extremal problems; many of these also have algorithmic counterparts.

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} .