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Conjecture Let 
be a simple 
-uniform hypergraph, and assume that every set of 
points is contained in at most 
edges. Then there exists an 
-edge-coloring so that any two edges which share vertices have distinct colors.
be a simple -uniform hypergraph, and assume that every set of points is contained in at most edges. Then there exists an -edge-coloring so that any two edges which share vertices have distinct colors. Vizing's Theorem is equivalent to the above statement for 
. 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.
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