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The following question came up at tea today, and none of us managed to come up with an answer. I was wondering if anyone had any ideas.

Does there exist a subset $X$ of $\mathbb{R}^2$ with the following two properties.

  1. If $p,q \in X$ are distinct, then the distance from $p$ to $q$ is at least $1$.
  2. There exists some $c \in \mathbb{R}$ such that if $R \subset \mathbb{R}^2 \setminus X$ is any closed rectangle (possibly "tilted", i.e. with its sides not necessarily parallel to the coordinate axes), then the area of $R$ is at most $c$.

Of course, $X$ must be infinite. As a weak guess, I would wager that no such $X$ exists, but I have no idea how to prove it.

EDIT : The rectangles in condition 2 include their interiors (so points in $X$ cannot occur in the interiors of the rectangles).

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    @AdamSmith: This appears to be a well-know open problem, see discussion at http://mathoverflow.net/questions/3307/can-a-discrete-set-of-the-plane-of-uniform-density-intersect-all-large-triangles.2014-04-04

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