mg.metric geometry - What is the max number of points in R^3, interconnected by generic curves?

Matt's answer is correct, but at an even simpler level: If you take two generic line segments in a compact subset of R^2, they'll intersect with positive probability. If you take two generic line segments in R^3, they'll intersect with probability 0. This isn't a proof by any means, but it's the simplest conceptual reason I know of. If instead of edges we wanted surfaces, we'd have to go up to dimension 5.

::sigh:: Okay, here's a constructive example of an infinite set of points such that no straight line segments between any two of them intersect. Take any two real numbers, say 2 and $pi$, that are algebraically independent. Then I claim that the set of points $(n, 2^n, pi^n)$ is such a set.

Why? Suppose the line segment between the points with $x = a$ and $x = b$ intersected the line segment between $x = c$ and $x = d$, parameterizing the line segments so the equations:

$(a, 2^a, pi^a) + lambda (b-a, 2^b - 2^a, pi^b - pi^a)$

$(c, 2^c, pi^c) + gamma (d-c, 2^d - 2^c, pi^d - pi^c)$

give us the same point for some choice of the variables.

Looking at the first two components tells us that $lambda, gamma$ are both rational. But then the third component gives us a polynomial with rational coefficients that has a root equal to $pi$, which is impossible since $pi$ is transcendental. So none of these line segments can intersect.

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