A note on reducing uniformity in Khot-Saket hypergraph coloring hardness reductions

In a recent result, Khot and Saket [FOCS 2014] proved the quasi-NP-hardness of coloring a 2-colorable 12-uniform hypergraph with 2(logn) Ω(1) colors. This result was proved using a novel outer PCP verifier which had a strong soundness guarantee. We reduce the arity in their result by modifying their 12-query inner verifier to an 8-query inner verifier based on the hypergraph coloring hardness reductions of Guruswami et al. [STOC 2014]. More precisely, we prove quasi-NP-hardness of the following problems on n-vertex hypergraphs. • Coloring a 2-colorable 8-uniform hypergraph with 2(logn) Ω(1) colors. • Coloring a 4-colorable 4-uniform hypergraph with 2(logn) Ω(1) colors.