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This paper studies visibility problems in Euclidean spaces $\mathbb{R}^d$where the obstacles are the points of infinite discrete sets$Y\subseteq\mathbb{R}^d$. A point $x\in\mathbb{R}^d$ is called$\varepsilon$-visible for $Y$ (notation: $x\in\mathbf{vis}(Y, \varepsilon))$ ifthere exists a ray $L\subseteq\mathbb{R}^d$ emanating from $x$ such that$||y-z||\geq\varepsilon$, for all $y\in Y\setminus\{x\}$ and $z\in L$. A point$x\in\mathbb{R}^d$ is called visible for $Y$ (notation: $x\in\mathbf{vis}(Y))$if $x\in\mathbf{vis}(Y, \varepsilon))$, for some $\varepsilon>0$.\\ Our mainresult is the following. For every $\varepsilon>0$ and every relatively denseset $Y\subseteq\mathbb{R}^2$, $\mathbf{vis}(Y, \varepsilon))\neq\mathbb{R}^2$.This result generalizes a theorem of Dumitrescu and Jiang, which settledMitchell's dark forest conjecture. On the other hand, we show that there existsa relatively dense subset $Y\subseteq \mathbb{Z}^d$ such that$\mathbf{vis}(Y)=\mathbb{R}^d$. (One easily verifies that$\mathbf{vis}(\mathbb{Z}^d)=\mathbb{R}^d\setminus\mathbb{Z}^d$, for all $d\geq2$). We derive a number of other results clarifying how the size of a sets$Y\subseteq\mathbb{R}^d$ may affect the sets $\mathbf{vis}(Y)$ and$\mathbf{vis}(Y,\varepsilon)$. We present a Ramsey type result concerninguniformly separated subsets of $\mathbb{R}^2$ whose growth is faster thanlinear.

On Visibility Problems with an Infinite Discrete, set of Obstacles