$\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}\newcommand{\R}{\mathbb{R}}\DeclareMathOperator{\E}{E}$

Erdős posed the following question, but it was asked to me by Ahmad Biniaz:

What is the maximum number of unit distances determined by a set of $n$ points in convex position?

Füredi was the first to prove an $O(n\log n)$ upper bound. Simpler proofs of the $O(n\log n)$ upper bound were later given by Braß and Pach and Braß, Károlyi and Valtr (alternate link).

Edelsbrunner gives a lower bound construction of a convex set of $n$ points that determines $2n-7$ unit distances. Even improving the constant in this lower bound would be a contribution.