# H5

For a finite set \(P\subset\mathbb R^2\), let
\[
\nu(P)=\#\bigl\{\{x,y\}\subset P:x\ne y,\ \|x-y\|_2=1\bigr\},
\]
where \(\|\cdot\|_2\) is the Euclidean norm. For every positive integer \(n\), define
\[
\nu(n)=\max\{\nu(P):P\subset\mathbb R^2,\ |P|=n\}.
\]
There exist a fixed real number \(\delta>0\) and a sequence of finite planar sets \((P_j)_{j\ge1}\) such that, writing \(n_j=|P_j|\),
\[
n_j\longrightarrow\infty,
\qquad
\nu(P_j)\ge n_j^{1+\delta}\quad\text{for every }j\ge1.
\]
Consequently, infinitely many distinct positive integers \(n\) satisfy
\[
\nu(n)\ge n^{1+\delta}.
\]
The same \(\delta\) applies to every set in the sequence.