Hypergraph regularity lemma

Statement

For 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.

• $\mathcal{P}$ is a $(\mu, \mathbf{\delta}, \mathbf{d}, r)$-equitable family of partitions and $a_i \leq T_0$ for every $i = 1, \ldots, k-1$.
• $\mathcal{H}^{(k)}$ is $(\delta_k, r)$ regular with respect to $\mathcal{P}$.