13,464 research outputs found
Type decomposition in NIP theories
We prove that any type in an NIP theory can be decomposed into a stable part
(a generically stable partial type) and a distal-like quotient.Comment: Several improvements made following the referee repor
A Note on "Regularity lemma for distal structures"
In a recent paper, Chernikov and Starchenko prove that graphs defined in
distal theories have strong regularity properties, generalizing previous
results about graphs defined by semi-algebraic relations. We give a shorter,
purely model-theoretic proof of this fact.Comment: 6 page
Finding generically stable measures
We discuss two constructions for obtaining generically stable Keisler
measures in an NIP theory. First, we show how to symmetrize an arbitrary
invariant measure to obtain a generically stable one from it. Next, we show
that suitable sigma-additive probability measures give rise to generically
stable measures. Also included is a proof that generically stable measures over
o-minimal theories and the p-adics are smooth
Dp-minimality: invariant types and dp-rank
This paper has two parts. In the first one, we prove that an invariant
dp-minimal type is either finitely satisfiable or definable. We also prove that
a definable version of the (p,q)-theorem holds in dp-minimal theories of small
or medium directionality. In the second part, we study dp-rank in dp-minimal
theories and show that it enjoys many nice properties. It is continuous,
definable in families and it can be characterised geometrically with no mention
of indiscernible sequences. In particular, if the structure expands a divisible
ordered abelian group, then dp-rank coincides with the dimension coming from
the order.Comment: New section added on dp-rank and the appendix with Sergei Starchenko
is now a separate pape
Invariant types in NIP theories
We study invariant types in NIP theories. Amongst other things: we prove a
definable version of the (p,q)-theorem in theories of small or medium
directionality; we construct a canonical retraction from the space of
M-invariant types to that of M-finitely satisfiable types; we show some
amalgamation results for invariant types and list a number of open questions.Comment: Small changes mad
VC-sets and generic compact domination
Let X be a closed subset of a locally compact second countable group G whose
family of translates has finite VC-dimension. We show that the topological
border of X has Haar measure 0. Under an extra technical hypothesis, this also
holds if X is constructible. We deduce from this generic compact domination for
definably amenable NIP groups.Comment: 15 page
NIP omega-categorical structures: the rank 1 case
We classify primitive, rank 1, omega-categorical structures having
polynomially many types over finite sets. For a fixed number of 4-types, we
show that there are only finitely many such structures and that all are built
out of finitely many linear orders interacting in a restricted number of ways.
As an example of application, we deduce the classification of primitive
structures homogeneous in a language consisting of n linear orders as well as
all reducts of such structures.Comment: Substantial changes made to the presentation, especially in sections
3 and
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