# Unit-distance H0-to-H5

## Setting

You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.

## Problem

Prove H5 below. No checkpoint result is granted.

**Target: 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.

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.