Partition Regularity of Pythagorean Triples (YEAR?)

Originators: R. Graham    (presented by J. Cooper - REGS 2009)

Definitions: An equation is partition-regular if every 2-coloring of the positive integers yields a solution to the equation using integers with the same color. Another way of saying this is that the hypergraph whose edges are the sets of integers forming solutions is not 2-colorable.

Question: Is the equation x²+y²=z² partition-regular? That is, does the hypergraph of Pythagorean triples fail to be 2-colorable?

Comments: There is a 2-coloring of the integers 1 through 1344 having no monochromatic Pythagorean triple. In general, one can color [n] using c log n colors and avoid monchromatic Pythagorean triples.