Note on the Pythagorean Triple System

We investigate some combinatorial aspects of the “Pythagorean triple system”. Our motivation is the following question: Is it possible to color the naturals with finitely many colors so that no Pythagorean triple is monochromatic? This question is open even for two colors. A natural approach is to search for a nonbipartite triple system that can be realized as a family of Pythagorean triples. Steiner triple systems (STS’s) are a rich source of such potential subconfigurations. However, we show that the Pythagorean triples – and, in fact, a large class of related triple systems – do not contain any STS. An equation is called “partition regular” if, for any coloring of the naturals (or integers) with finitely many colors, some (nontrivial) solution to the equation is monochromatic. For example, it is the first case of Van der Waerden’s famous Theorem that 2y = x + z is partition regular; this is another way to state that any coloring of N contains arithmetic progressions of length three in one color. Other celebrated results are Schur’s Theorem that x+ y = z is partition regular, and its broad generalization, Rado’s Theorem. Many more interesting examples and their relatives are discussed in [3]. Much less is known about the regularity of nonlinear equations. A very natural question is the regularity of the “Pythagorean equation” x+y = z. Is there a way to color N with finitely many colors so that no Pythagorean triple is monochromatic? It is not hard to see that there exists an O(log n)coloring of {1, . . . , n} (color k by the 5-adic order of k, i.e., the number of ∗University of South Carolina, Department of Mathematics, [email protected] †Virginia Tech, Department of Computer Science, [email protected]