In 1946, Paul Erdős posed the following question: What is the minimal number of distinct
distances determined by $n$ distinct points? We can ask a more refined question: Given a set
of $n - 1$ distinct distances $d_1,...,d_{n-1}$, can we arrange $n$ points in a way such that the
$d_1$ is determined exactly once, $d_2$ exactly twice, etc.? One solution, of course, is to
arrange the $n$ points such that they are collinear, with the same distance between any two
adjacent points. To find more interesting cases then, we impose the restriction that the $n$
points be in general position: We say $n$ points in $\mathbb{R}^d$ are in \textit{general
position} if no $d+1$ points lie in the same hyperplane and no $d+2$ points lie in the same
hypersphere. If $n$ points in general position determine a set of $n-1$ unique distances that
satisfy the previous requirement, then we say the $n$ points are in \textit{crescent
configuration}. In this talk, I will provide some history on the development built on Erdős's
original conjecture, and the near-optimal result proven by Guth \& Katz very recently in 2015.
But wait, there's more -- for small $n$ in low dimensions, can we actually construct and classify
all crescent configurations up to graph isomorphism? This problem is much more difficult than it
seems. I will talk about how we set up such constructions, and relevant recent developments.
(Much credit is due to my summer research advisor, Eyvindur Ari Palsson, whom I worked with
at the 2018 Park City Mathematics Institute.)