“Nelson asked: What is the smallest number of colors that you’d need to color any such graph, even one formed by linking an infinite number of vertices?”
The Hadwiger-Nelson problem is a problem that deals with the four color theorem and is it really true for infinite amount of vertices. Aubrey De Grey who is behind this says that it is not possible for four but it is for five colors. When he tried around 20,000 vertices it did not work with four. When he tried even just around 1,000 vertices it didn’t work with four either.