\documentclass[a4paper]{article}

\usepackage{amsmath}
\begin{document}

Recall that a complete first-order theory is VC-minimal if there are
$\phi_i(x,\bar{y})$ such that in any model $M$, any formula with parameters is
equivalent to a boolean combination of instances $\phi_i(x,\bar{b})$, and the
boolean subalgebra of $\mathcal{P}(M)$ generated by any pair
$(\phi_i(x,\bar{b}),\phi_j(x,\bar{c}))$ of instances is not free.

This notion was introduced by Adler as an abstraction of the ``swiss cheese''
decomposition admitted by algebraically closed valued fields, but as a
tameness notion it has some apparent deficiencies: the reduct of a VC-minimal
theory need not be VC-minimal, and VC-minimality has no known nice consequence
for the structure of definable sets in more than one variable. This paper
discusses three variants of VC-minimality, two of which are motivated by these
problems.

The first of these: $T$ is {\em convexly orderable} if some (any) model $M$ can be
equipped with a linear order such that any $T$-definable subset of $M$ is a finite
union of convex sets, with uniform bounds in definable families on the number
of convex sets. (It seems that, in contrast with the situation with strong
minimality and o-minimality, this explicit condition on uniformity of bounds
can not be removed.)

Convex orderability is clearly preserved under taking reducts. The authors
show that it is a strict weakening of VC-minimality, and that it implies that
formulae in one variable have vc-density $\leq 1$, and hence that it implies
dp-minimality.

Secondly: {\em weak VC-minimality} relaxes the condition in the above definition
of VC-minimality by restricting the assumption on pairs of instances to pairs
of instances of a common formula, $(\phi_i(x,\bar{b}),\phi_i(x,\bar{c}))$. The
authors show that this is a strict strengthening of NIP, and that in a weakly
VC-minimal theory any $\phi(x,\bar{y})$ with the order property has the strict
order property, and hence if the theory is unstable it interprets an infinite
linear order; they deduce that the Keuker conjecture holds for weakly
VC-minimal theories.

Lastly: {\em full VC-minimality} strengthens VC-minimality with a uniformity
assumption, namely by requiring in the above definition that any formula
without parameters is equivalent to a boolean combination of the
$\phi_i(x,\bar{y})$. Full VC-minimality is a very strong condition; as the authors
note, even $\operatorname{ACF}_0$ does not satisfy it (while any strongly
minimal theory is VC-minimal), although weak o-minimality does imply it.

The authors prove under this strong assumption a positive result about
definable sets in many variables - namely ``low VC-density'', meaning
that the VC-density of any $\phi(\bar{x},\bar{y})$ is bounded by the length of
$\bar{x}$. Whether this holds for VC-minimal theories remains an open question.

The paper concludes with some open questions.

The exposition throughout the paper is quite clear. Proofs are mostly
presented in great detail.

\end{document}
