 Hypergraph container theorem

For fixed uniformity, sufficiently small nonsingleton hypergraph codegrees relative to $d{\tau }^{|\sigma |-1}$ permit a family of containers whose degree measures are bounded below one by a fixed amount. Each container has a container fingerprint of size $O(\tau n)$. Iteration gives containers inducing at most an arbitrary fixed proportion of all hyperedges, with logarithmic family size $O(n\tau \log (1/\tau ))$. The constants depend on the uniformity and desired edge proportion, and the average degree must be positive.