Let $n$ be a natural number and let $S_n$ be a square $[0,n] \times [0,n]$ in the plane.
We say that a partition $\mathcal{Q} = R_1 \cup \cdots \cup R_t$ of $S_n$ is simple if each of the sets $R_1, \ldots, R_t$ is connected, has positive area and has diameter at most 1.
Question. Does there exist a collection of at most $k$ ($k$ does not depend on $n$) simple partitions $\mathcal{Q}_1, \ldots, \mathcal{Q}_k$ of $S_n$ such that any two points $x,y \in S_n$ of distance at most 1 are contained in at least one of the sets of the partitions, that is there exist $i \in\{1, \ldots, k\}$ and a set $R \in \mathcal{Q}_i$ such that $x,y \in R$?