Not every graph can be obtained as the intersection graph of, say, straight-line segments (or other geometric objects) in the plane. These graphs have many nice structural properties. In particular, they contain much larger homogeneous subgraphs than guaranteed by Ramsey's theorem. It seems that this phenomenon is related to some basic topological facts, including the Borsuk-Ulam theorem. But does it have anything to do with algebra? We discuss this question and some of its computational consequences. As a byproduct, we prove a conjecture of Erdos about distance distributions in 3-space.