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  "body": "\n## Definition\n\nLet $T$ be a [[complete theory|655K8Y7VS8]] in a first-order language $L$.\nLet $\\varphi(\\bar x; \\bar y)$ be an $L$-[[formula|HYJ10AF6RT]] whose free variables are split into two tuples $\\bar x$ and $\\bar y$.\nThe formula $\\varphi$ has the *independence property* in $T$ if the following holds for every $n \\in \\N$.\nThere are a [[model|MVCH6N9SNW]] $M$ of $T$, tuples $\\bar a_1, \\ldots, \\bar a_n$ in $M$, and tuples $\\bar b_J$ in $M$ for each $J \\subseteq \\set{1, \\ldots, n}$.\nFor all $i$ and $J$, $M \\models \\varphi(\\bar a_i; \\bar b_J)$ holds if and only if $i \\in J$.\nThe theory $T$ is *NIP* if no $L$-formula has the independence property in $T$.\n\n## Remarks\n\nExample: the theory of dense linear orders without endpoints is NIP.\nBy quantifier elimination, a formula $\\varphi(x; \\bar y)$ defines, for each parameter $\\bar b$, a union of at most $k$ points and intervals.\nThe endpoints are entries of $\\bar b$, and $k$ depends only on $\\varphi$.\nSuch a family cannot cut out all $2^n$ subsets of a set of $n$ points once $n$ is large.\nIt is a standard fact that it suffices to check formulas with $\\bar x$ a single variable.\n\nNon-example: the theory of the [[Rado graph|RMXR0FEF54]] is not NIP.\nFor the formula $x E y$, take distinct vertices $a_1, \\ldots, a_n$; by the extension property, for each $J$ there is a vertex $b_J$ adjacent to $a_i$ exactly when $i \\in J$.\n\nEquivalently, $T$ is NIP if for every formula $\\varphi(\\bar x; \\bar y)$ and every model $M$ of $T$, the family $\\set{\\varphi(M; \\bar b) : \\bar b \\in M}$ has finite [[VC dimension|KCFXR9K3GD]].\nEvery stable theory and every o-minimal theory is NIP.\n",
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