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Digraph extremal problems, hypergraph extremal problems, and the densities of graph structures. (English) Zbl 0541.05035
Let r and q be positive integers. An r-uniform directed q-hypergraph H is a set V(H) of vertices, together with a family E(H) of ordered r-tuples of distinct elements of V(H); an r-tuple with a given order may occur at most q times. We may also speak of an (r,q)-digraph. Let $$G^ m$$ be an (r,q)-digraph with m vertices. Let the vector $$u=(u_ 1,u_ 2,...,u_ m)$$ range over the standard (m-1)-simplex in $${\mathbb{R}}^ m$$. We consider the real multilinear form $$f_ G(u)=\sum u_{i_ 1}u_{i_ 2}...u_{i_ r}$$ summed over $$(i_ 1,i_ 2,...,i_ r)$$ such that $$(v_{i_ 1},v_{i_ 2},...,v_{i_ r})$$ is an edge of $$G^ m$$ with multiplicities taken into acount. The maximum of $$f_ G(u)$$ is called the density of $$G^ m$$ and denoted by $$g(G^ m)$$. For fixed r and q, the set of densities attained is denoted by $${\mathcal D}_ g$$. If ex(n,$${\mathbb{L}})$$ is the maximum number of oriented hyperedges in an n- vertex (r,q)-digraph not containing a member of $${\mathbb{L}}$$, $$\lim_{n\to \infty}ex(n,{\mathbb{L}})/n^ r$$ is called the extremal density of $${\mathbb{L}}$$. For fixed r and q, the set of extremal densities attained is denoted by $${\mathcal D}_ e$$. Motivated from results for ordinary graphs, digraphs, and multigraphs, relations between these two notions are established. For example, $${\mathcal D}_ g\subseteq {\mathcal D}_ e$$ and $${\mathcal D}_ g$$ is dense in $${\mathcal D}_ e$$.
Reviewer: K.-W.Lih

##### MSC:
 05C35 Extremal problems in graph theory 05C20 Directed graphs (digraphs), tournaments 05C65 Hypergraphs
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