Suslin Tree: Set Theory & Forcing

Updated 22 August 2025

A Suslin tree is an ω₁-tree with countable levels that lacks uncountable chains and antichains, serving as a fundamental combinatorial structure in set theory.
It plays a critical role in examining the independence of Suslin’s Hypothesis and in applying forcing techniques to explore the continuum and related topological spaces.
Variants such as full, homogeneous, and free Suslin trees provide diverse test cases for analyzing the interplay between combinatorics, Boolean algebras, and topological properties.

A Suslin tree is a well-founded combinatorial structure central to set theory, topology, and the theory of linear orders, especially in the context of independence phenomena related to the continuum and the countable chain condition. A Suslin tree is an ω₁-tree (height ω₁, countable levels) with no uncountable chains and no uncountable antichains; the existence of such a tree is independent of ZFC and equivalent to the failure of Suslin’s Hypothesis on linear orders. Suslin trees provide canonical counterexamples and test cases for the interaction between combinatorial set theory, forcing, and the structure theory of Boolean algebras, topological spaces, and ordered sets.

1. Formal Definition and Fundamental Properties

A Suslin tree is formally defined as a tree (T, <) satisfying:

Height(T) = ω₁;
Each level {x ∈ T: o(x) = α} is countable;
Every branch (maximal chain) is at most countable;
Every antichain is at most countable.

Here o(x) denotes the order-type of {y: y < x}. These trees are “tall” (height ω₁) but “narrow”: they avoid both uncountable branches and uncountable families of incomparable elements. Every Suslin tree is an Aronszajn tree, but not conversely: Aronszajn trees can have uncountable antichains.

Suslin trees are highly combinatorial objects, and under suitable set-theoretic assumptions (e.g., in Gödel’s constructible universe L), their existence can be ensured. Forcing techniques can also be used to introduce or destroy Suslin trees, illuminating their status as undecidable in ZFC (Smith et al., 21 Aug 2025).

2. Suslin Trees and Suslin’s Hypothesis

Suslin’s Hypothesis (SH) asks whether every dense complete linear order without endpoints that satisfies the countable chain condition is isomorphic to the real line (ℝ, <). A Suslin line is a dense, unbounded linear order with the c.c.c. that is not separable.

The equivalence between Suslin lines and Suslin trees is established as follows:

If a Suslin line exists, consider the tree T of nondegenerate closed intervals [a, b] of the line, ordered by reverse inclusion. The properties of the line translate to the Suslin properties for the associated tree.
Conversely, given a Suslin tree T, define the set S of all countable branches of T, ordered lexicographically. S is a dense linear order with the c.c.c. and is non-separable.

Thus, SH holds if and only if there is no Suslin tree, and the existence of a Suslin tree is equivalent to the failure of SH (Smith et al., 21 Aug 2025).

3. Forcing, Independence, and Model Theory

The existence of Suslin trees is independent of ZFC. Jensen proved that a Suslin tree exists in L, while Solovay, Tennenbaum, Martin, and others used forcing to construct models without Suslin trees. Shelah’s work (Brodsky et al., 2016) and subsequent developments have shown that even simple forcings, such as Cohen, Prikry, Magidor, and Radin forcing, can add a Suslin tree under mild cardinal arithmetic assumptions (e.g., a GCH-type hypothesis), via the introduction of certain club and coherence principles.

Forcing with a Suslin tree, or iterating with c.c.c. posets that preserve a fixed tree, permits fine-grained control over the existence of minimal uncountable linear orders (Soukup, 2018), special Aronszajn trees, and the topological properties of the continuum. In many results, special types of Suslin trees (e.g., coherent, free, or homogeneous) with enhanced combinatorial or rigidity features are constructed via elaborate forcing and combinatorial principles (Rinot et al., 2023, Krueger, 2022).

4. Combinatorial Structures and Variants

Variants of Suslin trees arise by imposing additional combinatorial properties:

Full Suslin tree: A κ-tree is full if at each limit level, at most one potential branch is omitted. Full κ-Souslin trees can exist at small cardinals under combinatorial principles such as the “proxy principle” or variants of Jensen’s diamond (Rinot et al., 2023).
Homogeneous Suslin tree: Every pair of nodes at the same level have isomorphic subtrees above them. Under strong square principles, homogeneous κ⁺-Souslin trees can be constructed (Rinot, 2011).
Free Suslin tree: All finite dimensional derived trees of the Suslin tree are themselves Suslin (n-free for all n < ω). Free Suslin trees can, by product forcing, be converted into highly rigid Kurepa trees with no Aronszajn subtree (Krueger, 2023).
Almost Kurepa Suslin trees: Suslin trees with the property that, after forcing with automorphisms, they “become” Kurepa trees (have ω₂-many cofinal branches), yet remain Suslin before forcing; such trees may lack a strongly saturated square, and their existence often relies on sophisticated forcing with Knaster or totally proper posets together with “strong separation” properties for automorphisms (Krueger et al., 31 Jan 2024, Krueger et al., 15 Jun 2024).
Coherent Suslin forests: Forests generalize trees, and coherent forests have the property that any two elements differ only on a small set. Coherent Suslin forests can be constructed by forcing using local approximations, variants of Cohen forcing, or strong diamond-like guessing principles, often starting from a large cardinal such as a Mahlo (Eskew, 2019).

A variety of additional qualitative notions have been introduced, such as entangledness and freeness (Krueger, 2019), and properties concerning saturation and non-saturation in products or squares of Suslin trees (Krueger et al., 15 Jun 2024).

5. Structural, Topological, and Descriptive Set-Theoretic Complexity

The class of Suslin trees is Π₁¹-complete in the projective hierarchy of subsets of the higher Baire space ω₁^{ω₁} under (V=L), and lacks the Baire property unless empty (Friedman et al., 2019). The complexity of the class is sensitive to set-theoretic context—for example, in models where every Aronszajn tree is special (e.g., via suitable forcing), there are no Suslin trees and the corresponding class is Borel.

Topologically, Suslin trees can behave unexpectedly under natural topologies: for example, there exist models in which an infinitely splitting Suslin tree is Lindelöf in the fine wedge topology, but its topological square fails to be Lindelöf, answering a longstanding question (Krueger, 9 Aug 2024).

6. Applications and Impact

Suslin trees have significant applications:

Set-Theoretic Topology: The structure of compact, normal, or paracompact spaces may depend on the existence of Suslin trees or coherent variants, which can be used to construct or eliminate pathological examples, as in the consistent resolution of the Arhangel’skiĭ–Tall problem by forcing with a coherent Souslin tree (Tall, 2011).
Partition Relations and Cardinal Invariants: The existence of a κ⁺-Souslin tree impacts partition relations—e.g., it implies that κ⁺ ↛ (κ⁺, log_κ(κ⁺) + 2)² (Raghavan et al., 2016).
Ordered Structures: The classification and minimality properties of linear orders, and the construction of rigid or homogeneous orders, often use Suslin trees as foundational tools (Smith et al., 21 Aug 2025, Krueger, 2019).
Boolean Algebras: Suslin trees produce examples of Boolean algebras with subtle properties related to the c.c.c., homogeneity, saturation, and completeness (Rinot, 2011, Eskew, 2019).
Inner Model Theory and Descriptive Set Theory: The notion of a Suslin cardinal, defined in terms of projective complexity or scales, is tied to the theory of cutpoints in mice; Suslin trees thus have deep connections to mouse limits and determinacy (Jackson et al., 2022).

7. Open Problems and Future Directions

Open questions include:

To what extent do combinatorial principles such as the proxy principle or diamond suffice for the construction of full or homogeneous Suslin trees at small cardinals without additional large cardinal assumptions (Rinot et al., 2023)?
The preservation of Suslinness in iterations and under various forcing notions, especially in connection with club isomorphisms, saturation, and automorphism structures (Krueger, 2022, Krueger et al., 31 Jan 2024, Krueger et al., 15 Jun 2024).
The exact behavior of the topological squares and higher power spaces associated to Suslin trees under specialized topologies, and their interaction with forcing (Krueger, 9 Aug 2024).
The possibility of constructing non-saturated Aronszajn trees in the absence of weak Kurepa trees and the further stratification of saturation and anti-chain phenomena (Krueger et al., 15 Jun 2024).

Advances in these areas continue to illuminate the interplay between combinatorics, topology, descriptive set theory, and the structure of the set-theoretic universe, with Suslin trees as a recurring central object of paper.