This problem appears in Croot and Lev's 2007 "Open Problems in Additive Combinatorics" (http://people.math.gatech.edu/~ecroot/E2S-01-11.pdf ), where it is attributed to Erdos and Graham (the latter of whom offers $250 for its solution).
Other references may include Cooper and Poirel's "Notes on the Pythagorean Triple System" (http://www.math.sc.edu/~cooper/pth.pdf ), which mentions that even the case of whether 2 colors is enough is open. They also exhibit a 2-coloring of the integers from 1 to 1344 without any monochromatic Pythagorean triples.
UPDATE (5/4/16): A new preprint of Heule, Kullmann, and Marek (to appear in the SAT 2016 conference) answers the question for $2$ subsets in the negative: If $\{1,2,\dots,7825\}$ is split into two subsets, one subset must contain a pythagorean triple. The $7825$ here is tight. The proof is heavily computer-assisted, and the key ideas seem to be to cast the problem in the language of satisfiability, and set up a divide-and-conquer method so that the (enormous) resulting instance is handleable by state-of-the-art satisfiability solvers.