On a combinatorial problem from the model theory of wreath products. II
Given integers \(r\), \(n\) with \(2\leq n\leq r\), denote by \(\delta(n,r)\) the least integer \(d\) for which there are disjoint multisets \(\{E_1,\dots, E_d\}\) and \(\{E_1',\dots, E_d'\}\) of equivalence relations on a set \(X\) of \(r\) elements such that each relation has at most \(n\) classes, at least one of them actually has \(n\) nonempty classes, and for each \(x\in X\) the restrictions of the \(E_j\) to \(X\setminus\{x\}\) coincide in some order with the restrictions of the \(E_j'\) to \(X\setminus\{x\}\). The numbers \(\delta(n,r)\) arise in the model-theoretic investigations of wreath products. The author gives the explicit values of \(\delta(n,r)\) for all arguments \(n\), \(r\). In Part I he proves the values for \(n=3\), in Part II those for \(n=4\).
model-theoretic investigations of wreath products
scientific article; zbMATH DE number 1295470