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  "body": "\n## Summary\n\nThe VC dimension (Vapnik–Chervonenkis dimension) of a [[family of sets|0C21Q694AG]] is the size of the largest set on which it cuts out every subset.\nIt measures the complexity of the family as a single number.\nBy the [[Sauer–Shelah lemma|Y4FQT5WPQJ]], a family of finite VC dimension $d$ picks out only polynomially many subsets, about $n^d$, of an $n$-point set.\nThis is why VC dimension controls sample sizes in statistical learning theory.\n\n## Explanations\n\n### L1 human\n\nThink of each set in the family as a rule that says \"yes\" to some points and \"no\" to others.\nA set of points is shattered when the rules together can produce every possible yes/no pattern on it.\nThe VC dimension is the largest number of points that can be shattered.\nFor example, intervals on the real line can shatter any two points but no three points $a \u003c b \u003c c$, because no interval contains $a$ and $c$ but not $b$.\nSo intervals have VC dimension $2$.\n",
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