Partition relations for trees I: Incomparable trees

Joint work with Tanmay Inamdar.

Abstract. Todorcevic proved that Martin’s axiom implies that every two coherent $\aleph_1$-Aronszajn trees are comparable. Here, from cardinal arithmetic assumptions, we obtain the failure of the analogous statement for higher trees. In particular, for every $n<\omega$, assuming $2^{\aleph_n}=\aleph_{n+1}$, there is a family of $\aleph_{n+2}$-many pairwise incomparable $\aleph_{n+1}$-coherent ${\aleph_{n+1}}$-complete $\aleph_{n+2}$-Aronszajn trees. The proof uses an anti-Ramsey colourings for trees recently introduced by the authors.

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