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  "body": "\n## Statement\n\nLet $X$ be a finite set with $n$ elements, let $d \\ge 0$ be an integer, and let $\\mathcal{F}$ be a [[family|0C21Q694AG]] of subsets of $X$.\nSuppose that $\\mathcal{F}$ has [[VC dimension|14HX3QF45H]] at most $d$, that is, no subset of $X$ with $d + 1$ elements is shattered by $\\mathcal{F}$.\nThen\n$$|\\mathcal{F}| \\le \\sum_{i=0}^{d} \\binom{n}{i}.$$\n\n## Remarks\n\nA stronger statement holds: every family $\\mathcal{F}$ shatters at least $|\\mathcal{F}|$ subsets of $X$, where the empty set counts and is shattered by every nonempty family.\nIt is proved by induction on $n$, splitting $\\mathcal{F}$ according to a fixed element $x$.\nIf the VC dimension is at most $d$, all shattered sets have at most $d$ elements, which gives the lemma.\n\nThe bound is sharp: for $d \\le n$, the family of all subsets of $X$ with at most $d$ elements has VC dimension $d$ and exactly $\\sum_{i \\le d} \\binom{n}{i}$ members.\nFor fixed $d$ the bound is a polynomial of degree $d$ in $n$, while $X$ has $2^n$ subsets.\n\n## Explanations\n\n### L1 human\n\nIf a family of sets cannot realise every pattern on any $d + 1$ points, then it is small: its size grows only like $n^d$ instead of $2^n$.\nSo a single local obstruction forces a strong global limit.\n\n### L2 human\n\nIn learning theory, the lemma bounds the number of ways a class of classifiers with finite [[VC dimension|KCFXR9K3GD]] can label $n$ points.\nThis polynomial growth is the key step in the proof that such classes are learnable from a number of samples that depends on the dimension.\nIn combinatorial geometry it bounds the complexity of set systems such as half-planes, which leads to the existence of small $\\varepsilon$-nets.\n",
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