  Given a set of $n$ points in $\mathbb R^d$, the (monochromatic) Closest Pair
problem asks to find a pair of distinct points in the set that are closest in
the $\ell_p$-metric. Closest Pair is a fundamental problem in Computational
Geometry and understanding its fine-grained complexity in the Euclidean metric
when $d=\omega(\log n)$ was raised as an open question in recent works
(Abboud-Rubinstein-Williams [FOCS'17], Williams [SODA'18],
David-Karthik-Laekhanukit [SoCG'18]).
  In this paper, we show that for every $p\in\mathbb R_{\ge 1}\cup\{0\}$, under
the Strong Exponential Time Hypothesis (SETH), for every $\varepsilon>0$, the
following holds:
  $\bullet$ No algorithm running in time $O(n^{2-\varepsilon})$ can solve the
Closest Pair problem in $d=(\log n)^{\Omega_{\varepsilon}(1)}$ dimensions in
the $\ell_p$-metric.
  $\bullet$ There exists $\delta = \delta(\varepsilon)>0$ and $c =
c(\varepsilon)\ge 1$ such that no algorithm running in time
$O(n^{1.5-\varepsilon})$ can approximate Closest Pair problem to a factor of
$(1+\delta)$ in $d\ge c\log n$ dimensions in the $\ell_p$-metric.
  At the heart of all our proofs is the construction of a dense bipartite graph
with low contact dimension, i.e., we construct a balanced bipartite graph on
$n$ vertices with $n^{2-\varepsilon}$ edges whose vertices can be realized as
points in a $(\log n)^{\Omega_\varepsilon(1)}$-dimensional Euclidean space such
that every pair of vertices which have an edge in the graph are at distance
exactly 1 and every other pair of vertices are at distance greater than 1. This
graph construction is inspired by the construction of locally dense codes
introduced by Dumer-Miccancio-Sudan [IEEE Trans. Inf. Theory'03].
