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  "body": "\n## Statement\n\nFor all positive real $\\mu$, $\\delta_k$, and functions $r \\, \\colon \\mathbb{N} \\times (0, 1]^{k-2} \\to \\mathbb{N}$, $\\delta_j \\, \\colon (0,1]^{k-j} \\to (0,1]$ for $j = 2, \\ldots, k-1$ there exists $T_0$ and $n_0$ so that the following holds. For any $k$-uniform hypergraph $\\mathcal{H}^{(k)}$ on $n \\geq n_0$ vertices, there exists a family of partitions $\\mathcal{P} = \\mathcal{P}(k-1, \\mathbf{a})$ and a vector $\\mathbf{d} = (d_2, \\ldots, d_{k-1})$ so that, for $r = r(a_1, \\mathbf{d})$ and $\\mathbf{\\delta} = (\\delta_2, \\ldots, \\delta_{k-1})$ where $\\delta_j = \\delta_j(d_j, \\ldots, d_{k-1})$ for all $j$, the following holds.\n\n- $\\mathcal{P}$ is a $(\\mu, \\mathbf{\\delta}, \\mathbf{d}, r)$-equitable family of partitions and $a_i \\leq T_0$ for every $i = 1, \\ldots, k-1$.\n- $\\mathcal{H}^{(k)}$ is $(\\delta_k, r)$ regular with respect to $\\mathcal{P}$.\n",
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        "locator": "Hypergraph regularity method, section Statements \u003e Hypergraph regularity lemma",
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  "name": "Hypergraph regularity lemma",
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  "role": "lemma",
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