Terry has a nice post up bout the Erdös-Ulam problem, which was unfamiliar to me. Here’s the problem:

Let S be a subset of R^2 such that the distance between any two points in S is a rational number. Can we conclude that S is not topologically dense?

S doesn’t have to be finite; one could have S be the set of rational points on a line, for instance. But this appears to be almost the only screwy case. One can ask, more ambitiously:

Is it the case that there exists a curve X of degree <= 2 containing all but 4 points of S?

Terry explains in his post how to show something like this conditional on the Bombieri-Lang conjecture. The idea: lay down 4 points in general position. Then the condition that the 5th point has rational distances from x1,x2,x3, and x4 means that point lifts to a rational point on…

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