We are given an arbitrary finite subset $P$ of the plane containing $N$ points. Let $Q$ be a subset of $P$ such that the pairwise distances $d(p,q)$ are unique for all $p,q\in Q$. 


EDIT: How large can $Q$ be while remaining a strict subset of $P$? Is there a non-trivial lower bound for $Q$?



The size of a subset $Q$ of $P\subset\mathbb{R}^2$ that has all distinct pairwise distances
An earlier version of this answer had a simple induction proof that $|Q|$ is bounded below by a nonconstant function of $|P|$ for points in any dimension,
but I have since discovered that much stronger bounds are known.


For some of the history of the problem, and generalizations to volumes of higher-dimensional simplexes, see "[Applications of the Canonical Ramsey Theorem to Geometry](https://arxiv.org/abs/1302.5334)", Gasarch and Zbarsky, arXiv:1302.5334. According to them, the best bound known for your two-dimensional problem is that every set of $n$ points in 2d has a distinct-distance subset of size $\Omega(n^{1/3}/\log n)$. The reference for this result is "[A note on distinct distance subsets](https://arxiv.org/abs/1211.1776)", Charalambides, *J. Geom.* 2013.


