Maximal stable quotients of invariant types in NIP theories

Abstract

For a NIP theory $T$, a sufficiently saturated model $\mathfrak{C}$ of $T$, and an invariant (over some small subset of $\mathfrak{C}$) global type $p$, we prove that there exists a finest relatively type-definable over a small set of parameters from $\mathfrak{C}$ equivalence relation on the set of realizations of $p$ which has stable quotient. This is a counterpart for equivalence relations of the main result of the paper "On maximal stable quotients of definable groups in NIP theories" by M. Haskel and A. Pillay which shows the existence of maximal stable quotients of type-definable groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in the paper "On first order amenability" by E. Hrushovski, K. Krupinski, and A. Pillay.