define a graph g by taking the vertices as $\mathbb{r}^2$ and the edges to be any pair of vertices that are distance 1 apart. the hadwiger-nelson problem asks the chromatic number of this graph, written $\chi(\mathbb{r}^2)$. it is known that either $4\leq \chi(\mathbb{r}^2)\leq 7$ or $5\leq \chi(\mathbb{r}^2)\leq 7$. we explore some approaches to solving this problem, encountering along the way the axiom of choice (or lack thereof) and other infinite oddities.