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 Kriz (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：我们证明共识划分定理蕴含了Kneser超图色数的下界，为Alon、Frankl与Lovász（《美国数学会会刊》，1986年）的结论及其由Kriz（《美国数学会会刊》，1992年）推广的结果提供了新颖证明。该方法被应用于研究全搜索问题Kneser$^p$的计算复杂性——该问题在给定一个以少于色数的颜色对$p$一致Kneser超图进行着色的简洁表示时，要求寻找一个单色超边。我们证明：对每个素数$p$，通过扩展方式访问输入着色的Kneser$^p$问题可高效归约至含有$p$份的共识划分问题的某个相当弱近似。特别地，当$p=2$时，该问题可高效归约至归一化单调函数共识平分问题的任意非平凡近似。我们进一步证明：对每个素数$p$，Kneser$^p$问题属于复杂性类$\mathsf{PPA}$-$p$。作为应用，我们确立了受限于有界颜色数着色的Kneser$^p$问题的复杂性上界。