A few years ago, the Boolean Pythagorean Triples problem was solved and shown to be impossible for $n\ge 7825$. Is there a way to show that the number $n$ is bounded just with Ramsey theory, and if so, could anyone point me to the proper reference? All the articles I seem to find deal with SAT solvers as opposed to the Ramsey theory behind the problem.
2026-03-25 19:00:02.1774465202
Boolean Pythagorean Triples Problem
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No such proof is currently known.
As a toy example, it is easy to show that it's impossible to color the integers with three colors such that every Pythagorean triple includes all three colors: take the four triples $$(319, 360, 481), (31, 480, 481), (481, 600, 769), (360, 480, 600)$$ which are impossible to color in this way. (Say the colors are red, blue, and green, and $481$ is red. Then the first three triples ensure that none of $360$, $480$, or $600$ can be red; therefore $(360, 480, 600)$ is colored only using blue and green.)
There was some work done even prior to 2016 about trying to find a more complicated, but still small configuration like this one for the two-color problem. For example, the Fano plane is impossible to two-color without getting a monochromatic line, so if we could find a Fano plane in the Pythagorean triples hypergraph, we'd obtain a short proof. However, Cooper and Poirel proved in 2008 that the Fano plane is not realizable via Pythagorean triples, and in fact, neither is any Steiner triple system.
In principle, it's still possible that some set of only a few (likely very large) triples can be found that's not two-colorable, and this would give a human-readable proof of the result. However, searching for other small configurations among the Pythagorean triples is hard, and very similar to still-open problems; for example, the perfect cuboid problem.