MAXIMAL STABLE QUOTIENTS OF INVARIANT TYPES IN NIP THEORIES

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 [2] on 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 [3].