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On November 19th, 2009 Andrew King says:

Haxell proved that $ (2.75 + \epsilon)\Delta $ is sufficient for sufficiently large $ \Delta $ depending on $ \epsilon $. (JGT, 2008)

Aharoni, Berger, and Ziv proved the fractional relaxation, i.e. that with partition cliques of size $ 2\Delta $ we have a fractional $ 2\Delta $ colouring. (Combinatorica, 2007)

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