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  "body": "\n## Statement\n\nLet $A$ and $B$ be finite sets, let $f : A \\to B$ be a function, and let $n$ be a natural number.\nAssume that $|B| \\cdot n \u003c |A|$.\nThen there is an element $b \\in B$ whose fibre $f^{-1}(b) = \\{a \\in A : f(a) = b\\}$ has more than $n$ elements.\n\n## Remarks\n\nFor $n = 1$ this is the [[pigeonhole principle|0HGVCWQGY3]].\nAn equivalent form: if $N$ objects go into $k$ boxes, some box has at least $\\lceil N / k \\rceil$ objects.\nExample: among $25$ people, at least $3$ share a birth month, because $12 \\cdot 2 \u003c 25$.\n\n## Explanations\n\n### L1 human\n\nIf every box held at most $n$ objects, the boxes would hold at most $n$ times the number of boxes.\nSo if there are more objects than that, some box must hold more than $n$.\n",
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  "front": {
    "kind": "statement",
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    "name": "Generalized pigeonhole principle",
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        "locator": "Strong form",
        "url": "https://en.wikipedia.org/wiki/Pigeonhole_principle"
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  "markdown": "https://math2.tamnd.com/node/0BY7GN3449.md",
  "msc": [
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  "name": "Generalized pigeonhole principle",
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  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0BY/0BY7GN3449.md",
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  "tag": "0BY7GN3449",
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  "url": "https://math2.tamnd.com/wiki/generalized-pigeonhole-principle/",
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    "tag": "0BY7GN3449",
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