Re:Mathematicians Bridge Finite-Infinite Divide
by SilasX on Wednesday May 25, 2016 @03:46PM
(#11772517)
But it's not true for any coloring, e.g. "red if a and b < 20, blue otherwise".
Re:Mathematicians Bridge Finite-Infinite Divide
by NotAPerson on Wednesday May 25, 2016 @04:03PM
(#11772656)
Wouldn't the subset {(a, b) | a,b > 20} be monochromatic?
Ed:
Perhaps I phrased it poorly, but I think the point was to show that you can always construct a predicate, P over a and b, such that P(a, b) is finitely defined (such as "a > 20 and b > 20"), but {(a, b) | P(a, b) is true} is infinite and monochromatic.
Instead of having some cases of colorings where your only option is to construct things of the form "(a = 5 and b = 17) or (a = 3 and b = 47) or ..." where you just list out every pair that matches (in an infinite subset).
I'm pretty sure that the challenge is to prove that you can for any coloring construct a finitely defined rule for picking the members of the subset which is guaranteed to give you a monochromatic subset.
The question is more about if you can always find such a (finite) rule to partition the set, rather than if you can in a few easily constructed examples.