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Source. Dániel T. Soukup, Trees, ladders and graphs, J. Combin. Theory Ser. B 115 (2015), 96--116, doi:10.1016/j.jctb.2015.05.004; Theorem 4.3 on p. 10 of arXiv:1409.2922v1, the edition read and identified on the source card. Labels and pages are those of arXiv v1.

Statement

The tree $T(S)$, its comparability graph $G(T)$ and separation by a set of vertices are as on the Theorem 3.5 page; $<_T$ is the tree order.

Theorem 4.3 (p. 10, quoted). "Fix a stationary, costationary $S\subset\omega_1$ and let $T=T(S)$. Then there is a subgraph $X$ of $G(T)$ such that $Chr(X)=\omega_1$ and any two $<_T$-incomparable points are separated by a finite set in $X$. In particular, every uncountable set $A\subseteq T$ contains two vertices which are separated by a finite set in $X$."

Here the separating set works for all paths of $X$, not only for paths inside a given vertex set; the paper notes that this is stronger than the absence of uncountable $\omega$-connected sets in Theorem 3.5 (p. 9). The introduction describes the result as a graph in which every uncountable set contains two points joined by only finitely many pairwise disjoint paths, even in the whole graph (p. 2). In the proof the separating set for incomparable $t,t'$ is a finite set of predecessors of $t$ (Lemma 4.2, p. 10), so it contains neither point.

Read depth. Claims checked: the statement, Definition 4.1 and Lemma 4.2 were read clause by clause on the printed pages. The proof (pp. 10--15) was not checked.

Proof pointer

Definition 4.1 (p. 9) calls a ladder system coherent when $C_s=C_t\cap s^\downarrow$ whenever $t$ is in its support (infinite ladders), $s\in C_t$ and $C_s$ is finite, and there is a true ladder system $\underline\eta$ (one-point ladders at successors, cofinal $\omega$-sequences at limits) compatible with its infinite ladders. Lemma 4.2 (p. 10) shows that for a tree with no branching at limits and a transitive coherent ladder system $\underline C$, any two $<_T$-incomparable points are separated by a finite set in $X_{\underline C}$. The proof of Theorem 4.3 (pp. 10--15) builds such a system on $T(S)$ with $\operatorname{Chr}(X_{\underline C})>\omega$, by an induction over levels like that of Theorem 3.5 with coherence witnessed by a true ladder system induced from one on $\omega_1$; Claim 4.3.1 (p. 13) checks transitivity and coherence, and Claim 4.3.2 (pp. 14--15) supplies the colouring argument. Every uncountable subset of $T$ contains two incomparable points because $T$ has no uncountable chains.

Dependencies

Theorem 3.5 and its Lemma 3.3, with Definition 4.1 and Lemma 4.2 of the same paper.

Bears on

• Problem 1067: Theorem 4.3 gives a second negative answer of the same kind as Theorem 3.5: every uncountable vertex set of $X$ contains two vertices joined by only finitely many disjoint paths of $X$, so no subgraph of $X$ of chromatic number $\aleph_1$ is infinitely connected.