Worked solution: Independence of Suslin's Hypothesis from ZFC (Jensen, Solovay–Tennenbaum, 1971)

Step 2 of 8: Trees translate the question into pure combinatorics

In plain words

Picture a family tree: each person has a well-defined set of ancestors, generations count height, an infinite unbroken line of descent is a branch, and a group of people none of whom is an ancestor of another is an antichain. An " $\omega_1$ -tree" grows for $\aleph_1$ -many uncountable generations.

A Suslin tree is a tree of this height that manages the surprising trick of having no uncountably long branch and no uncountably large antichain, even though it has $\aleph_1$ -many nodes overall — a strange combinatorial object that turns out to exist exactly when a Suslin line does.

T \text{ tree}, \ \mathrm{ht}(T) = \omega_1, \ \text{no uncountable branch or antichain}

Detailed analysis

A tree is a partially ordered set $(T, <_T)$ in which the set of predecessors of any node is well-ordered by $<_T$ ; the height of a node is the order type of its predecessors, and the height of $T$ is the supremum of the heights of its nodes. A branch is a maximal linearly ordered subset (a "chain" running through the tree), and an antichain is a set of pairwise incomparable nodes.

An $\omega_1$ -Suslin tree is a tree of height $\omega_1$ in which every branch and every antichain is countable, yet the tree itself has $\aleph_1$ nodes. A classical equivalence (attributed to Kurepa's early work on the subject) shows that a Suslin line exists if and only if an $\omega_1$ -Suslin tree exists: given a Suslin line one builds a tree of intervals ordered by reverse inclusion, and given a Suslin tree one orders its branches lexicographically to build a line.

This equivalence is what lets set theorists attack Suslin's Hypothesis by pure transfinite recursion on trees rather than delicate order-theoretic arguments about continua — exactly the combinatorial machinery the next two steps put to use.

Terms in this step

Tree (order-theoretic): A partially ordered set in which the predecessors of every element are well-ordered; height, branches, and antichains are defined exactly as for a family tree of ancestry.
Branch and antichain: A branch is a maximal chain (totally ordered subset) running through a tree; an antichain is a set of pairwise incomparable nodes. A Suslin tree keeps both kinds of set countable despite having $\aleph_1$ nodes overall.