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  "body": "\n## Summary\n\nA NIP theory is a complete first-order theory in which no formula has the independence property.\nA formula has the independence property when, with suitable parameters, it can cut out every subset of arbitrarily large finite sets.\nNIP theories include both the [[stable theories|T7JW5QY6KR]] and the [[o-minimal theories|DTWHAW1N0V]].\nThe condition says that every definable family of sets has finite [[VC dimension|KCFXR9K3GD]], which links model theory with learning theory and combinatorics.\n\n## Explanations\n\n### L1 human\n\nFix a formula and think of it as a way to choose subsets, one subset for each value of the parameter.\nA formula is \"wild\" if, for every $n$, there are $n$ points such that every one of the $2^n$ subsets of them can be chosen.\nNIP means that no formula is wild in this way.\nOn the real line with its order, the formula $x \u003c y$ only chooses initial segments, so among two points $a \u003c b$ it can never choose $\\set{b}$ alone.\nIn the random graph, the edge relation can choose every subset of any finite set of vertices, so its theory is not NIP.\n",
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