We prove that coloring a 3-uniform 2-colorable hypergraph with c colors is NP-hard for any constant c. The best known algorithm  colors such a graph using O(n 1/5) colors. Our result immediately implies that for any constants k ≥ 3 and c 2 > c 1 > 1, coloring a k-uniform c 1-colorable hypergraph with c 2 colors is NP-hard; the case k = 2, however, remains wide open.
This is the first hardness result for approximately-coloring a 3-uniform hypergraph that is colorable with a constant number of colors. For k ≥ 4 such a result has been shown by , who also discussed the inherent difference between the k = 3 case and k ≥ 4.
Our proof presents a new connection between the Long-Code and the Kneser graph, and relies on the high chromatic numbers of the Kneser graph [19,22] and the Schrijver graph . We prove a certain maximization variant of the Kneser conjecture, namely that any coloring of the Kneser graph by fewer colors than its chromatic number, has ‘many’ non-monochromatic edges.
Mathematics Subject Classification (2000):68Q17
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