Question 110
Consider that we have omitted some of the points of a 2d-lattice (same as $\mathbb{Z}^2$). Look at these points as a graph, two points (vertices) are connected if they are the same in one of their coordinates and differ 1 in another. Each connected component is called a *cluster*. Suppose that for all $n \in \mathbb{N}$, the number of omitted points inside the horizontal square with side length $2n + 1$ and the origin as its center, be less than $n/2$. Prove that the non-omitted points contain exactly one infinite cluster.