- Published 2008

We comment on a question of Justin Moore on colourings of pairs of nodes in an Aronszajn tree. Lemma 1.1. For any uncountable antichain A in an Aronszajn tree T , there is an infinite chain C in T such that every element of C is the meet of two elements of A. Proof. Define a sequence 〈(ti, ui, vi, Bi) : i < ω〉 by recursion as follows. To start let t0 be an arbitrary element of A (note that t0 cannot be the root of T ). Consider all t0 ∧ a for a ∈ A and notice that this is a countable set. Hence there is u0 <T t0 such that {a ∈ A : u0 = a ∧ t0} is uncountable. Note that u0 has at least two distinct immediate successors in T , so let v0 be an immediate successor of u0 which is incompatible with t0 and which has uncountably many extensions in A. Denote the set of extensions of v0 in A by B0. At the stage i = j+1 we choose ti ∈ Bj and ui <T ti such that {a ∈ Bj : ui = a∧ ti} is uncountable. Choose v0 to be an immediate successor of ui which is incompatible with ti and which has uncountably many extensions in Bj . Let this set of extensions be Bi. Note that uj <T vj <T ui <T ti. At the end the set {ui : i < ω} is an infinite chain such that ui = ti ∧ ti+1. ⋆1.1 Definition 1.2. (1) A subtree of an ω1-tree T will mean an uncountable meet closed subset of T . (2) A subtree S of an ω1-tree is binary if every node of S has at most two distinct immediate successors in S. (3) An ω1-tree T is binarisable if every subtree of T has a binary subtree. (4) For a tree T we let T [2] = {{s, t} : s <T t}. The above definition will be used in the context of Aronszajn trees, so the case of trivial binary trees, namely uncountable branches, will be avoided. We will use the following statement introduced in [1] and used in [3]: Colouring Axiom for Trees (CAT): For any partition T = K0 ∪K1 of an Aronszajn tree T , there is an uncountable set X ⊆ T and i < 2 such that x ∧ y ∈ Ki for all distinct x, y ∈ X.

@inproceedings{Leffler200815A,
title={1 5 A ug 2 00 6 An application of CAT Mirna},
author={Mittag - Leffler},
year={2008}
}