Consider the graph whose vertex set is every point in the real plane. We will say two vertices in this graph are adjacent whenever they are exactly unit distance apart. For example, the points $(0, 0)$ and $(\sqrt{2}/2, \sqrt{2}/2)$ are adjacent in this graph.
The Hadwiger-Nelson Problem asks for the chromatic number of the graph defined above. It turns out it is not too hard to show the chromatic number at least 4 and no more than 7 (the Wikipedia article discusses these bounds), but it gets very strange to go much further. For example, it has been shown the answer may vary depending on which model of set theory you work in!