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  "body": "\n## Problem\n\nFor each integer $n \\ge 0$, let $M(n)$ be the [[Dedekind number|481ZPQ0GC3]], the number of monotone Boolean functions of $n$ variables.\nFind the value of $M(n)$ for every $n$, ideally by a closed formula.\n\n## Remarks\n\nNo closed formula of a practical kind is known, and the problem is open in this form.\nExact values are known only for $n \\le 9$; the value $M(9)$ was computed in 2023 with large computer searches.\nSimple bounds come from antichains: every family of $\\floor{n/2}$-element subsets of $\\set{1, \\ldots, n}$ is an antichain, so $M(n) \\ge 2^{\\binom{n}{\\floor{n/2}}}$.\nKleitman proved that $\\log_2 M(n) = (1 + o(1)) \\binom{n}{\\floor{n/2}}$ as $n \\to \\infty$, and finer asymptotic formulas are known.\n\n## Explanations\n\n### L1 human\n\nThe question is how many monotone yes/no rules on $n$ switches exist.\nFor a few switches one can count by hand: $2, 3, 6, 20, 168$ for $n = 0, 1, 2, 3, 4$.\nThe counts explode so quickly that even with computers only the first ten values are known.\nA formula that gives them all is still missing.\n",
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      "first_stated": "1897",
      "posed_by": "Richard Dedekind"
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  "msc": [
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  "name": "Dedekind's problem",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/RM6/RM64J2CP1M.md",
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