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Motivated by the Erd\H{o}s-Faber-Lov\'asz (EFL) conjecture for hypergraphs,we consider the list edge coloring of linear hypergraphs. We discuss severalconjectures for list edge coloring linear hypergraphs that generalize both EFLand Vizing's theorem for graphs. For example, we conjecture that in a linearhypergraph of rank 3, the list edge chromatic number is at most 2 times themaximum degree plus 1. We show that for sufficiently large fixed rank andsufficiently large degree, the conjectures are true.

Linear Hypergraph List Edge Coloring - Generalizations of the EFL Conjecture to List Coloring