The hardness of 3-uniform hypergraph coloring

We prove that coloring a 3-uniform 2-colorable hypergraph with c colors is NP-hard for any constant c. The best known algorithm [20] 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 [14], 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 [26]. 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.