Mathematics – Combinatorics
Scientific paper
2008-09-20
Mathematics
Combinatorics
8 pages, 2 figures
Scientific paper
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).
Cooper Joshua
Poirel Chris
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