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.

翻译：我们证明了共识分割定理可用于推导Kneser超图色数的下界，从而为Alon、Frankl和Lovász（Trans. Amer. Math. Soc., 1986）的经典结论及其由Kříž（Trans. Amer. Math. Soc., 1992）提出的推广提供了新颖证明。我们将此方法应用于研究完全搜索问题Kneser$^p$的计算复杂性：给定一个$p$-均匀Kneser超图的简洁着色表示（所用颜色数少于其色数），要求找出单色超边。我们证明对于任意素数$p$，若允许扩展访问输入着色信息，则Kneser$^p$问题可高效归约为$p$份额共识分割问题的弱近似求解。特别地，当$p=2$时，该问题可高效归约为归一化单调函数上共识平分问题的任意非平凡近似求解。进一步地，我们证明对于任意素数$p$，Kneser$^p$问题属于复杂度类$\mathsf{PPA}$-$p$。作为应用，我们针对颜色数有界的着色情形，建立了Kneser$^p$问题复杂性的局限性。