= Auxiliary matching construction for hypergraph edge colouring
Given an $r$-uniform <hypergraph> $H$ and $K$ colours, construct an $(r+1)$-uniform auxiliary <hypergraph> with one task vertex for every $e\in E(H)$ and one resource vertex $(v,c)$ for every original vertex and colour. The assignment edge for $(e,c)$ contains the task $e$ and all resources $(v,c)$ with $v\in e$. A <matching in a hypergraph> of assignments is exactly a proper partial edge colouring. Task degrees are $K$; resource degrees are $d_H(v)$. Pairwise auxiliary <hypergraph codegrees> are at most $\max(1,\max_{v\ne w}d_H(v,w))$. Thus nearly equal original degrees and small original <hypergraph codegrees> transfer to the setting of a <Rödl nibble>. A locally balanced <Rödl nibble> must control the unmatched task count in every original star, not merely the total uncovered proportion.
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