Primitive bound of a 2-structure

Abstract

A 2-structure on a set $S$ is given by an equivalence relation on the set of ordered pairs of distinct elements of $S$. A subset $C$ of $S$, any two elements of which appear the same from the perspective of each element of the complement of $C$, is called a clan. The number of elements that must be added in order to obtain a 2-structure the only clans of which are trivial is called the primitive bound of the 2-structure. The primitive bound is determined for arbitrary 2-structures of any cardinality. This generalizes the classical results of Erd\H{o}s et al. and Moon for tournaments, as well as the result of Brignall et al. for finite graphs, and the precise results of Boussa\"{\i}ri and Ille for finite graphs, providing new proofs which avoid extensive use of induction in the finite case.