{
  "statement": "Prove H5 below. No checkpoint result is granted.\n\n**Target: H5**\n\nFor a finite set \\(P\\subset\\mathbb R^2\\), let\n\\[\n\\nu(P)=\\#\\bigl\\{\\{x,y\\}\\subset P:x\\ne y,\\ \\|x-y\\|_2=1\\bigr\\},\n\\]\nwhere \\(\\|\\cdot\\|_2\\) is the Euclidean norm. For every positive integer \\(n\\), define\n\\[\n\\nu(n)=\\max\\{\\nu(P):P\\subset\\mathbb R^2,\\ |P|=n\\}.\n\\]\nThere 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\\[\nn_j\\longrightarrow\\infty,\n\\qquad\n\\nu(P_j)\\ge n_j^{1+\\delta}\\quad\\text{for every }j\\ge1.\n\\]\nConsequently, infinitely many distinct positive integers \\(n\\) satisfy\n\\[\n\\nu(n)\\ge n^{1+\\delta}.\n\\]\nThe same \\(\\delta\\) applies to every set in the sequence.",
  "starting_statement": null,
  "target_statement": "For a finite set \\(P\\subset\\mathbb R^2\\), let\n\\[\n\\nu(P)=\\#\\bigl\\{\\{x,y\\}\\subset P:x\\ne y,\\ \\|x-y\\|_2=1\\bigr\\},\n\\]\nwhere \\(\\|\\cdot\\|_2\\) is the Euclidean norm. For every positive integer \\(n\\), define\n\\[\n\\nu(n)=\\max\\{\\nu(P):P\\subset\\mathbb R^2,\\ |P|=n\\}.\n\\]\nThere 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\\[\nn_j\\longrightarrow\\infty,\n\\qquad\n\\nu(P_j)\\ge n_j^{1+\\delta}\\quad\\text{for every }j\\ge1.\n\\]\nConsequently, infinitely many distinct positive integers \\(n\\) satisfy\n\\[\n\\nu(n)\\ge n^{1+\\delta}.\n\\]\nThe same \\(\\delta\\) applies to every set in the sequence.",
  "case_id": "H0-to-H5",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}