On the existence of indiscernible trees

Abstract

We introduce several concepts concerning the indiscernibility of trees. A tree is by definition an ordered set (O, < ini, where η< iniν means that η is a proper initial segment of ν. In this paper, we consider some structure M in the language L and are interested in sets A of the form (aη)η∈O, where O is a tree, and a η labeled by η is an element in M. Such a set A is also called a tree in this paper. We study the indiscernibility of trees A in general settings and apply the obtained results to the study of unstable theories. © 2012 Elsevier B.V.