The Chromatic Number of Kneser Hypergraphs via Consensus Division

Abstract

We show that the Consensus Division theorem implies lower bounds on the chromatic number of Kneser hypergraphs, offering a novel proof for a result of Alon, Frankl, and Lov\'{a}sz (Trans. Amer. Math. Soc., 1986) and for its generalization by K\v{r}\'{\i}\v{z} (Trans. Amer. Math. Soc., 1992). Our approach is applied to study the computational complexity of the total search problem Kneser$^p$, which given a succinct representation of a coloring of a $p$-uniform Kneser hypergraph with fewer colors than its chromatic number, asks to find a monochromatic hyperedge. We prove that for every prime $p$, the Kneser$^p$ problem with an extended access to the input coloring is efficiently reducible to a quite weak approximation of the Consensus Division problem with $p$ shares. In particular, for $p=2$, the problem is efficiently reducible to any non-trivial approximation of the Consensus Halving problem on normalized monotone functions. We further show that for every prime $p$, the Kneser$^p$ problem lies in the complexity class $\mathsf{PPA}$-$p$. As an application, we establish limitations on the complexity of the Kneser$^p$ problem, restricted to colorings with a bounded number of colors.