We study the kernel complexity of constraint satisfaction problems over a finite domain, parameterized by the number of variables, whose constraint language consists of two relations: the non-equality relation and an additional permutation-invariant relation $R$. We establish a conditional lower bound on the kernel size in terms of the largest arity of an OR relation definable from $R$. Building on this, we investigate the kernel complexity of uniformly rainbow free coloring problems. In these problems, for fixed positive integers $d$, $\ell$, and $q \geq d$, we are given a graph $G$ on $n$ vertices and a collection $\cal F$ of $\ell$-tuples of $d$-subsets of its vertex set, and the goal is to decide whether there exists a proper coloring of $G$ with $q$ colors such that no $\ell$-tuple in $\cal F$ is uniformly rainbow, that is, no tuple has all its sets colored with the same $d$ distinct colors. We determine, for all admissible values of $d$, $\ell$, and $q$, the infimum over all values $η$ for which the problem admits a kernel of size $O(n^η)$, under the assumption $\mathsf{NP} \nsubseteq \mathsf{coNP/poly}$. As applications, we obtain nearly tight bounds on the kernel complexity of various coloring problems under diverse settings and parameterizations. This includes graph coloring problems parameterized by the vertex-deletion distance to a disjoint union of cliques, resolving a question of Schalken (2020), as well as uniform hypergraph coloring problems parameterized by the number of vertices, extending results of Jansen and Pieterse (2019) and Beukers (2021).

翻译：我们研究了有限域上约束满足问题的核复杂度，该问题以变量数量为参数化，其约束语言包含两种关系：非相等关系和一个额外的置换不变关系$R$。我们根据从$R$可定义的OR关系的最大元数，建立了核大小的条件下界。在此基础上，我们研究了均匀彩虹自由着色问题的核复杂度。在这类问题中，给定固定正整数$d$、$\ell$和$q \geq d$，我们有一个$n$个顶点的图$G$和一个由顶点集的$d$子集构成的$\ell$元组集合$\cal F$，目标是判定是否存在一个用$q$种颜色对$G$进行正常着色的方式，使得$\cal F$中没有$\ell$元组是均匀彩虹的，即没有元组的所有子集都被赋予相同的$d$种不同颜色。在假设$\mathsf{NP} \nsubseteq \mathsf{coNP/poly}$的前提下，我们确定了对于所有可允许的$d$、$\ell$、$q$值，使得该问题存在$O(n^η)$大小核的$η$的下确界。作为应用，我们在多种不同设置和参数化下，获得了各种着色问题核复杂度的近似紧界。这包括以顶点删除距离到团不交并作为参数化的图着色问题（解决了Schalken（2020）的一个问题），以及以顶点数量为参数化的均匀超图着色问题（扩展了Jansen和Pieterse（2019）以及Beukers（2021）的结果）。