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 $x,y \in R$?

strictly less than 1are contained in a single element of one of the partitions. I think the answer to that is no for similar reasons to those given by Gerhard Paseman. If, on the other hand, you ask that for a fixed $a<1$, that any two points $x$ and $y$ with $d(x,y)\le a$ belong to a single element of one of the partitions, the answer should be yes. $\endgroup$ – Anthony Quas Dec 1 '15 at 20:28