{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\n\nLet $n \\ge 2$, and let $\\norm{p - q}$ be the [[Euclidean distance|0NPPGHQNXQ]] in the [[Euclidean plane|0XW7V9ZZ05]].\nLet $u(n)$ be the largest number of pairs $\\set{p, q}$ with $\\norm{p - q} = 1$ in a set of $n$ points of the plane.\nDetermine the order of growth of $u(n)$ as $n \\to \\infty$.\n\n## Remarks\n\nPaul Erdős posed the problem in 1946.\nIn the same paper he showed $u(n) = O(n^{3/2})$, and, using a suitably scaled square grid, $u(n) \\ge n^{1 + c/\\log \\log n}$ for a constant $c \u003e 0$; here $O$ is [[big O notation|0RYTGBJAGN]].\nJoel Spencer, Endre Szemerédi and William Trotter proved $u(n) = O(n^{4/3})$ in 1984.\nA short proof of this bound by László Székely uses the [[crossing number inequality|34B1PW5EE2]].\nThe exact order of growth is still not known.\nThe pairs at distance $1$ form the edges of a unit distance graph on the $n$ points.\n\n## Explanations\n\n### L1 human\n\nPlace $n$ points in the plane and count the pairs that are exactly one unit apart.\nHow large can this count be?\nA square grid of the right size does a little better than $n$, and nobody has done much better.\nThe best upper bound is about $n^{4/3}$, and closing the gap is a famous open question of Erdős.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "0XW7V9ZZ05"
    ],
    "history": {
      "first_stated": "1946",
      "posed_by": "Paul Erdős"
    },
    "kind": "problem",
    "msc": [
      "52C10",
      "05C62"
    ],
    "name": "Unit distance problem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Unit_distance_graph"
      }
    ],
    "tag": "EPWB2JEXPE",
    "uses": [
      "0XW7V9ZZ05",
      "0NPPGHQNXQ",
      "0RYTGBJAGN",
      "34B1PW5EE2"
    ]
  },
  "kind": "problem",
  "link": "https://tamnd.github.io/math2/node/EPWB2JEXPE/",
  "links": [
    {
      "field": "about",
      "target": "0XW7V9ZZ05"
    },
    {
      "field": "uses",
      "target": "0XW7V9ZZ05"
    },
    {
      "field": "uses",
      "target": "0NPPGHQNXQ"
    },
    {
      "field": "uses",
      "target": "0RYTGBJAGN"
    },
    {
      "field": "uses",
      "target": "34B1PW5EE2"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/EPWB2JEXPE.md",
  "msc": [
    "52C10",
    "05C62"
  ],
  "name": "Unit distance problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EPW/EPWB2JEXPE.md",
  "status": null,
  "tag": "EPWB2JEXPE",
  "uri": "m2://EPWB2JEXPE",
  "url": "https://tamnd.github.io/math2/wiki/unit-distance-problem/",
  "writing": {
    "tag": "EPWB2JEXPE",
    "kind": "problem",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}