Extremal broadcasting problems

Extremal Problems for Geometric Hypergraphs 1 Extremal Problems for Geometric

A geometric hypergraph H is a collection of i-dimensional simplices, called hyperedges or, simply, edges, induced by some (i + 1)-tuples of a vertex set V in general position in d-space. The topological structure of geometric graphs, i.e., the case d = 2; i = 1, has been studied extensively, and it proved to be instrumental for the solution of a wide range of problems in combinatorial and compu...

Extremal Graph Problems, Degenerate Extremal Problems, and Supersaturated Graphs

Notation. Given a graph, hypergraph Gn, . . . , the upper index always denotes the number of vertices, e(G), v(G) and χ(G) denote the number of edges, vertices and the chromatic number of G respectively. Given a family L of graphs, hypergraphs, ex(n,L) denotes the maximum number of edges (hyperedges) a graph (hypergraph)Gn of order n can have without containing subgraphs (subhypergraphs) from L...

Higher Level Extremal Problems

Multicoloured extremal problems

Many problems in extremal set theory can be formulated as finding the largest set system (or r-uniform set system) on a fixed ground set X that does not contain some forbidden configuration of sets. We shall consider multicoloured versions of such problems, defined as follows. Given a list of set systems, which we think of as colours, we call another set system multicoloured if for each of its ...

On Multigraph Extremal Problems

1 . Introduction . In this paper we shall consider multigraphs and digraphs (= directed graphs) with bounded multiplicity : an integer r is fixed and we shall assume, that the considered multigraphs or digraphs have no loops, further, if u and v are two vertices of a multigraph M, they can be joined by more than one edge, however, they cannot be joined by more than r edges. In case of digraphs ...