Se p 20 08 Pythagorean Partition - Regularity and Ordered Triple Systems with the Sum Property Joshua Cooper

Is it possible to color the naturals with finitely many colors so that no Pythagorean triple is monochromatic? This question is even open for two colors. A natural strategy is to show that some small nonbipartite triple systems cannot be realized as a family of Pythagorean triples. It suffices to consider partial triple systems (PTS’s), and it is therefore natural to consider the Fano plane, the smallest nonbipartite PTS. We show that the Pythagorean triples do not contain any Fano plane. In fact, our main result is that a much larger family of “ordered” triple systems (viz. those with a certain “sum property”) do not contain any Steiner triple system (STS). An equation over the integers is called “partition regular” if, for any coloring of the naturals (or integers) with finitely many colors, some solution to the equation is monochromatic. For example, it is the first nontrivial 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 offshoots 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. University of South Carolina, Department of Mathematics, [email protected] Virginia Tech, Department of Computer Science, [email protected]