First and Second Order Signatures of Extreme Uniform Hypergraphs and Their Relationship with Vectors of the Vertex Degrees

Finding Set Extreme 3-Uniform Hypergraphs Cardinality through Second-Order Signatures

“…In [39], the «signature» for EUH 2 n is presented. In [56], an extension of the notion of signature is presented: a second-order signature, or vector of signatures, for EUH 3 n . A brief description of the basis is given in the introduction.…”

“…In [56], we also introduced the notion of a truncated signature and defined the relation of including one signature into another through the notion of truncation.…”

“…Each signature at one level restricts the possible signatures at the next level. For example, as shown in the [56] for σ = 25, the signatures 0, 1, 2, 2, 3, 4, 5, 8, 9 will be included. And the signature σ = 2 p − 1 includes all signatures from 0 to 2 p−1 − 1.…”

“…In [56,57], a new representation is given for an extremal 3-uniform hypergraph, and operations under signatures are described.…”

“…The structure of the base makes it quite easy to enumerate all its possible variants but makes it almost a simple enumeration. Thanks to the second-order signature representation, which appeared in the works of [56,57], as well as operations on these representations of extreme 3 uniform hypergraphs, we are able to propose a new, more efficient algorithm for calculating the cardinality of a set.…”