Are monochromatic Pythagorean triples avoidable?

A Pythagorean triple is a triple of positive integers a,b,c ∈ N+ satisfying a2 + b2 = c2. Is it true that, for any finite coloring of N+, at least one Pythagorean triple must be monochromatic? In other words, is the Diophantine equation X2 +Y 2 = Z2 regular? This problem has been open since several decades, even restricted to 2-colorings. In this note, we introduce partial morphisms, which are special colorings of N+ by finite groups with partly multiplicative properties. We show that, for many such colorings, monochromatic Pythagorean triples are impossible to avoid in the integer interval [1,10000].