{"schema":"vela.problem-packet.v0.1","problem":956,"statement":"If $C,D\\subseteq \\mathbb{R}^2$ then the distance between $C$ and $D$ is defined by\\[\\delta(C,D)=\\inf_{\\substack{c\\in C\\\\ d\\in D}}\\| c-d\\|.\\]Let $h(n)$ be the maximal number of unit distances between disjoint convex translates. That is, the maximal $m$ such that there is a compact convex set $C\\subset \\mathbb{R}^2$ and a set $X$ of size $n$ such that all $(C+x)_{x\\in X}$ are disjoint and there are $m$ pairs $x_1,x_2\\in X$ such that\\[\\delta(C+x_1,C+x_2)=1.\\]Determine $h(n)$ - in particular, prove that there exists a constant $c>0$ such that $h(n)>n^{1+c}$ for all large $n$.","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}