An inverse theorem in ${\mathbb{F}}_p$ and rainbow free colorings

Abstract

Let ${\mathbb{F}}_p$ be the field with $p$ elements with $p$ prime, $X_1,\dots ,X_n$ pairwise disjoint subsets of ${\mathbb{F}}_p$with at least $3$ elements such that ${\sum }_{i=1}^n|X_i|\le p-5$, and ${\mathbb{S}}_n$ the set of permutations of $\{1,2,\dots ,n\}$. If $a_1,\dots ,a_n\in {\mathbb{F}}_p^{\ast }$ are not all equal, we characterize the subsets $X_1,\dots ,X_n$ which satisfy \begin{equation*} \Bigg|\bigcup_{σ\in\mathbb{S}_n}\sum_{i=1}^na_{σ(i)}X_i\Bigg|\leq \sum_{i=1}^n|X_i|. \end{equation*} This result has the following application: For $n\ge 2$, $b\in {\mathbb{F}}_p$ and $a_1,\dots ,a_n$ as above, we characterize the colorings ${\bigcup }_{i=1}^n{C}_i={\mathbb{F}}_p$ where each color class has at least 3 elements such that ${\sum }_{i=1}^n{a}_i{x}_i=b$ has not rainbow solutions.

References in corpus (1)

• The structure of rainbow-free colorings for linear equations on three variables in Zp