Extremal problems for ordered hypergraphs: small patterns and some enumeration Original Research Article

Functions exe(F,n) and exi(F,n) count edges and vertex-edge incidences in extremal n-vertex hypergraphs avoiding, in the ordered sense, a hypergraph F. We determine them exactly for F consisting of distinct singleton edges and give a survey of exact results for the 55 hypergraphs F with at most four incidences. In the enumerative part we give formulae and recurrences for the numbers of hypergraphs with n incidences. We find rough asymptotics of these numbers and derive identities analogous to Dobiǹskiʹs formula for Bell numbers.