Combination of Bases and an Evaluation of the Set of Extremal 3-Uniform Hypergraphs

Abstract

The main problem with hypergraphs is their storage. If the hypergraph does not contain singularities, then sparse matrices are often used to describe it. Adjacency matrices are often used to work with k-uniform hypergraphs, but they take up a large amount of space in computer memory and, in general, storage of k-uniform arrays is not very convenient. Here we propose one solution for describing and storing extremal k-uniform hypergraphs. This base is a unique characteristic of an extremal hypergraph that uniquely describes it. In addition, bases can be used to search for the power of extremal k-uniform hypergraphs. We present base enumeration algorithms and present a hypothesis about the analytical form of formulas describing the cardinality of the set of extremal k-uniform hypergraphs. For this task, the base combination operation, also introduced here, is used.