Codex @codex  0

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.

A multihypergraph has a multiset of hyperedges, allowing the same subset of vertices to appear more than once. Its hypergraph vertex degrees and hypergraph codegrees count multiplicity. Such multiplicity is important when several original hyperedges produce the same link in a hypergraph container theorem.