Fractional hypergraph coloring

We investigate proper ( a : b ) -fractional colorings of n -uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned b distinct colors from a set of a colors, and an edge is properly colored if no single color is shared by all vertices of the edge. A hypergraph is ( a : b ) -colorable if every edge is properly colored. We prove that for any 2 ≤ b ≤ a − 2 and a ≤ n / ( 2 ln n ) , every n -uniform hypergraph H with | E ( H ) | ≤ ( a b 3 ) − 1 / 2 n ln n 1 / 2 a b n − 1 is ( a : b ) -colorable. We also give a corresponding probabilistic upper bound and discuss specific cases.