A generalization of Vizing's Theorem?
Status open low confidence
The Rosenfeld conjecture — that a simple $d$-uniform hypergraph in which every $(d-1)$-set is contained in at most $r$ edges admits an $(r+d-1)$-edge-coloring with no two edges sharing $d-1$ vertices receiving the same color — appears to remain open. Extensive searching found no verified paper proving or disproving it; the problem was noted as hard already at posting time because Kempe-chain arguments (the main tool for Vizing's theorem) fail for $d\geq 3$. Related but distinct Vizing-type generalizations to linear hypergraphs (the Berge–Füredi conjecture) have been studied actively, but these concern a different intersection regime.
Reviewer notes. The OPG page (openproblemgarden.org and garden.irmacs.sfu.ca) was inaccessible (ECONNREFUSED), so any post-2007 comments on that page could not be read. A 2016 Discrete Applied Mathematics paper 'Edge-coloring of 3-uniform hypergraphs' (DOI: 10.1016/j.dam.2016.07.003) appeared in several searches and may be relevant, but its full text was unreadable (binary PDF) so it could not be verified as addressing the Rosenfeld conjecture specifically — it was therefore excluded. The Berge–Füredi conjecture (for linear hypergraphs, q(H) ≤ Δ([H]₂)+1) is a different but related Vizing generalization and has received activity in 2024 (arXiv:2403.06850, arXiv:2403.09518); it is not the same as the Rosenfeld conjecture. Confidence is low due to inaccessibility of the OPG page and inability to read several candidate PDFs.
Discussion
Vizing's Theorem is equivalent to the above statement for $ d=2 $ . For higher dimensions, this problem looks difficult since the main tool used in the proof of Vizing's theorem (Kempe chains) do not appear to work.