What are the best results for upper bounds on the number of colours required in a *strong vertex colouring* of a regular hypergraph H?

A regular hypergraph is one in which every vertex is contained in k edges, for some constant k. (The edges may contain more than two vertices, and may contain different numbers of vertices from each other.)

A strong vertex colouring is one in which, for each edge, every vertex contained in that edge has a different colour.

I am hoping for an upper bound formulated in terms of the degree k of the vertices, the maximum cardinality of any edge, and other graph parameters — but without imposing any restrictions on the hypergraphs, aside possibly from a bound on edge cardinality. I would be especially interested in constructive proofs (*i.e.* ones which describe algorithms, or at least randomized constructions with high probability of success).

[**Note.** This question originally asked about edge-chromatic numbers in uniform hypergraphs, which is an equivalent problem. I have substantially shortened this question, and rephrased it in the form above, in the hopes that I might answers using a different presentation.]

randomuniform hypergraphs is known, in a (long!) paper by Krivelevich and Sudakov. $\endgroup$