# Small sets in dense pairs

## TL;DR

This paper proves that in certain dense pairs of o-minimal structures, the induced structure on the dense set eliminates imaginaries, allowing small definable sets to be embedded into finite Cartesian powers of the dense set.

## Contribution

It establishes conditions under which the induced structure on a dense set eliminates imaginaries in dense pairs of o-minimal structures, addressing a previously open question.

## Key findings

- Elimination of imaginaries in the induced structure on P
- Small definable sets can be embedded into P^l
- Verification of tameness conditions in key examples

## Abstract

Let $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ be an expansion of an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, such that three tameness conditions hold. We prove that the induced structure on $P$ by $\mathcal M$ eliminates imaginaries. As a corollary, we obtain that every small set $X$ definable in $\widetilde{\mathcal M}$ can be definably embedded into some $P^l$, uniformly in parameters, settling a question from [10]. We verify the tameness conditions in three examples: dense pairs of real closed fields, expansions of $\mathcal M$ by a dense independent set, and expansions by a dense divisible multiplicative group with the Mann property. Along the way, we point out a gap in the proof of a relevant elimination of imaginaries result in Wencel [17]. The above results are in contrast to recent literature, as it is known in general that $\widetilde{\mathcal M}$ does not eliminate imaginaries, and neither it nor the induced structure on $P$ admits definable Skolem functions.