Almost Kurepa Suslin trees and destructibility of the Guessing Model Property

This paper establishes the consistency of the Guessing Model Principle at $\omega_2$ alongside the existence of an almost Kurepa Suslin tree, demonstrating that the principle can be destructible by a ccc forcing of size $\omega_1$, while also proving the consistency of a weak Kurepa tree existing with the failure of the Kurepa Hypothesis and a specific guessing model principle that implies the tree property at $\omega_2$.

The Gist

Imagine the universe of mathematics as a vast, infinite city built on a foundation of logic. In this city, there are structures called trees. But these aren't trees with leaves and roots; they are mathematical structures where every "branch" goes up, and at every level, there are choices to be made.

Some of these trees are very well-behaved. Others are chaotic. Mathematicians have spent decades trying to understand which trees can exist and which cannot, because the answer tells us a lot about the fundamental rules of the universe.

This paper, written by Chris Lambie-Hanson and Šárka Stejskalová, is like a report from two architects who have just built two very strange, new neighborhoods in this city. They are proving that you can have a set of rules that usually keeps the city safe and orderly, but under specific conditions, those rules can be broken by a tiny, seemingly harmless construction crew.

Here is the breakdown of their discovery using everyday analogies.

1. The Rules of the City: "Guessing Models"

First, let's talk about the Guessing Model Property (GMP).

Imagine a super-intelligent city planner (a "Guessing Model") who can look at any small neighborhood in the city and perfectly predict what the whole city looks like. If this planner exists, the city is incredibly stable. It means there are no "rogue" structures (like infinite trees with too many branches) that could cause chaos.

In the world of large numbers (infinite cardinals), if this "Guessing Model" exists, the city is very robust. It's like having a force field that prevents bad things from happening.

2. The "Almost Kurepa" Tree: The Trojan Horse

The authors introduce a specific type of tree called an Almost Kurepa Suslin Tree.

• The Suslin Tree: Imagine a tree that is perfectly balanced. It has infinite height, but at any point, it has only a manageable number of branches. It's a "good" tree.
• The Kurepa Tree: This is a "bad" tree. It's so tall and wide that it has too many branches (more than the city can handle). If a Kurepa tree exists, the city's rules break down.
• The "Almost" Kurepa Tree: This is the trickster. In the current city, it looks like a perfect, well-behaved Suslin tree. It has no extra branches. However, it is a "Trojan Horse." If you force the city to accept a new rule (a mathematical process called "forcing") that adds just one new branch to this tree, the tree instantly explodes. Suddenly, it has too many branches, and the city's rules (the Guessing Model) collapse.

The Discovery: The authors proved that you can build a city where the "Guessing Model" (the super-planner) is active and working perfectly, BUT there is also this Trojan Horse tree sitting in the middle of town.

3. The Big Surprise: Fragility

The most exciting part of their paper is the destructibility.

For a long time, mathematicians thought that if you had a city with a "Guessing Model," it would be unbreakable. They thought no matter what small construction crew you sent in, the city would stay safe.

The authors showed this is false.

• They built a model where the Guessing Model is active.
• They showed that a very small construction crew (a "ccc forcing of size $\omega_1$"—think of it as a tiny team of workers) can come in, activate the Trojan Horse tree, and instantly destroy the Guessing Model.

The Analogy: It's like having a castle with an impenetrable shield. You assume it's safe from a single arrow. But the authors found a specific type of arrow (the Almost Kurepa tree) that, if fired, doesn't just pierce the shield; it causes the entire shield to shatter.

4. The Second Discovery: Separating the Twins

The paper also tackles a second problem involving Weak Kurepa Trees.

Imagine two twins:

1. The Kurepa Hypothesis (KH): "There exists a tree with too many branches."
2. The Weak Kurepa Hypothesis (wKH): "There exists a slightly less crazy tree with too many branches."

Usually, if the "Weak" version is false (meaning no crazy trees exist), the "Strong" version is also false. The authors wanted to know: Can we have a city where the "Weak" version is false (no crazy trees), but the "Strong" version is also false, yet we still have a "Weak Kurepa" tree?

Wait, that sounds confusing. Let's simplify:
They proved it is possible to have a city where:

• The "Guessing Model" is working (mostly).
• There are no Kurepa trees (the really bad ones).
• BUT there is a Weak Kurepa tree (a slightly less bad one).

This separates the two concepts. It shows that the rules preventing the "really bad" trees are different from the rules preventing the "slightly bad" trees. You can ban one without banning the other.

Why Does This Matter?

In the real world of mathematics, these trees represent the limits of what is possible.

• Robustness: For a long time, we thought some mathematical rules were "indestructible." This paper shows they are fragile.
• Flexibility: It shows that the universe of math is more flexible than we thought. You can mix and match these "tree" properties in ways that were previously thought impossible.
• The "Almost" Concept: The idea that something can look perfect now but be a disaster waiting to happen (the Almost Kurepa tree) is a powerful new tool for mathematicians to build new models of reality.

Summary

Think of this paper as a story about a city that thought it was safe.

1. The city had a "Super Planner" (Guessing Model) that kept everything orderly.
2. The authors found a "Hidden Trap" (Almost Kurepa Tree) that looks safe but destroys the Super Planner if you touch it.
3. They proved you can build a city with the Super Planner and the Trap, meaning the city's safety is fragile and can be broken by a tiny, simple action.
4. They also showed you can have a city where the "Super Bad" trees are banned, but the "Mildly Bad" trees are allowed, proving these two bans are independent.

It's a discovery that shakes the foundations of how we think about stability in the infinite world of mathematics.

Technical Summary

Here is a detailed technical summary of the paper "Almost Kurepa Suslin Trees and Destructibility of the Guessing Model Property" by Chris Lambie-Hanson and Šárka Stejskalová.

1. Problem Statement and Context

The paper addresses fundamental questions regarding the robustness (indestructibility) of compactness principles at accessible cardinals, specifically at $\omega_2$, under forcing extensions.

• The Tree Property ($TP(\kappa)$): Asserts that every $\kappa$-tree has a cofinal branch. At inaccessible cardinals, this characterizes weakly compact cardinals. At $\omega_2$, it is equiconsistent with a weakly compact cardinal.
• The Guessing Model Property ($GMP$): A strengthening of the tree property that characterizes supercompact cardinals among inaccessibles. $GMP(\omega_2)$ (denoted simply as $GMP$) implies the failure of the Kurepa Hypothesis ($\neg KH$) and the Weak Kurepa Hypothesis ($\neg wKH$).
• The Core Question: While compactness principles at large cardinals are often robust under "nice" forcings (like ccc or $\mu$-cc forcings for $\mu < \kappa$), it is unknown whether $TP(\omega_2)$ or $GMP$ are robust at $\omega_2$. Specifically:
  • Is it consistent that $GMP$ holds but is destroyed by a ccc forcing of size $\omega_1$?
  • Can one have $\neg wKH$ (no weak Kurepa trees) while simultaneously having an "almost Kurepa" Suslin tree (a Suslin tree that becomes Kurepa after forcing with itself)?

2. Methodology

The authors employ advanced forcing techniques, primarily variations of Mitchell forcing, combined with the theory of tree automorphisms and guessing models.

A. Modified Mitchell Forcing (Theorem A)

To prove the consistency of $GMP$ alongside an almost Kurepa Suslin tree, the authors construct a forcing extension using a variation of the classical Mitchell forcing $\mathcal{M}(\omega, \kappa)$.

• Standard Mitchell Forcing: Iterates $\text{Add}(\omega, 1)$ (adding Cohen subsets) at inaccessible stages to collapse cardinals and force $2^\omega = \kappa = \omega_2$.
• The Variation: Instead of adding Cohen subsets, the iterands add generic automorphisms to a fixed free Suslin tree $T$.
  • Let $A(T)$ be the forcing poset of finite approximations of automorphisms of $T$.
  • The forcing $\mathcal{M}^T_\kappa$ consists of pairs $(p, q)$ where $p$ adds Cohen reals and $q$ is a countable support function assigning $A(T)$-names to automorphisms at specific stages.
• Key Technical Challenge: Unlike the classical Mitchell forcing where the term forcing is $\omega_1$-closed, here the term forcing is not obviously $\omega_1$-closed. The authors develop a sophisticated $\omega_1$-approximation property argument (using elementary submodels and density arguments on derived trees) to show that the forcing preserves $\omega_1$ and maintains the necessary structural properties of the tree $T$.

B. Product Forcing (Theorem B)

To separate the Weak Kurepa Hypothesis from the Kurepa Hypothesis, the authors use a product of:

1. The classical Mitchell forcing $\mathcal{M}(\omega, \kappa)$ (to ensure $GMP_{\omega_2}(\omega_2)$ and $\neg KH$).
2. A specific forcing $\mathcal{K}_\kappa$ designed to add a weak Kurepa tree (a tree of height $\omega_1$ with $\kappa$ branches).

• They prove that $\mathcal{K}_\kappa$ is $\mu$-Knaster for appropriate $\mu$ and that the product preserves the necessary compactness properties while introducing the weak Kurepa tree.

C. Preservation Lemmas

The paper utilizes and extends preservation lemmas regarding:

• $\omega_1$-approximation property: Crucial for ensuring that no new cofinal branches are added to trees of height $\omega_1$ by the "tail" of the iteration.
• Distributivity: Showing that certain quotients of the forcing do not add new branches to trees.
• $\sigma$-centered forcing: A direct proof is provided that $\neg wKH$ is preserved by any $\sigma$-centered forcing (a class including Cohen forcing).

3. Key Contributions and Results

Theorem A: Destructibility of $GMP$

Assumption: Existence of a supercompact cardinal $\kappa$.
Result: There exists a forcing extension where:

1. $GMP$ holds.
2. There exists an almost Kurepa Suslin tree $T$.
   • $T$ is a Suslin tree in the ground model.
   • Forcing with $T$ adds a cofinal branch, making $T$ a Kurepa tree in the extension.
   • Since $GMP$ implies $\neg KH$ (no Kurepa trees), forcing with $T$ destroys $GMP$.
     Significance: This proves that $GMP$ is not indestructible by all ccc forcings of size $\omega_1$. It is consistent that $GMP$ holds but is destroyed by a ccc forcing of cardinality $\omega_1$. This contrasts with the fact that $GMP$ is indestructible by adding Cohen reals (countable support products).

Corollary 1.1: Assuming only an inaccessible cardinal, it is consistent that $\neg wKH$ holds but is destructible by a ccc forcing of size $\omega_1$.

Theorem B: Separation of Kurepa Hypotheses

Assumption: Existence of a supercompact cardinal $\kappa$.
Result: There exists a forcing extension where:

1. $2^\omega = \omega_2 = \kappa$.
2. $GMP_{\omega_2}(\omega_2)$ holds. (This implies the Tree Property at $\omega_2$).
3. $\neg KH$ holds (No Kurepa trees).
4. $wKH$ holds (There exists a weak Kurepa tree).
   Significance: This refines previous results by separating the Weak Kurepa Hypothesis from the Kurepa Hypothesis in the presence of strong compactness principles. It shows that $GMP_{\omega_2}(\omega_2)$ does not imply $\neg wKH$, whereas full $GMP$ (at $\omega_2$) does.

Theorem 5.2: Preservation of $\neg wKH$

Result: The failure of the Weak Kurepa Hypothesis ($\neg wKH$) is preserved by any $\sigma$-centered forcing.
Significance: This provides a direct proof (independent of the Guessing Model Property) that adding Cohen reals (or any $\sigma$-centered forcing) cannot create a weak Kurepa tree if one did not exist before. This highlights a sharp distinction: while $GMP$ can be destroyed by ccc forcing of size $\omega_1$, the weaker consequence $\neg wKH$ is robust against $\sigma$-centered forcings.

4. Significance and Implications

1. Sharpness of Indestructibility: The paper demonstrates that the robustness of $GMP$ is delicate. While it survives Cohen forcing, it can be destroyed by a ccc forcing of size $\omega_1$ if an almost Kurepa Suslin tree is present. This settles a question about the limits of indestructibility for compactness principles at $\omega_2$.
2. Combinatorial Structure of Suslin Trees: The construction of the almost Kurepa Suslin tree relies on the existence of $\omega_2$-many pairwise almost disjoint automorphisms. This connects the existence of such trees to the non-saturation of Aronszajn trees ($S \otimes S$ is non-saturated).
3. Hierarchy of Compactness: The results clarify the hierarchy between $GMP$, $GMP_{\omega_2}(\omega_2)$, $TP(\omega_2)$, and the various Kurepa hypotheses. Specifically, it shows that $GMP_{\omega_2}(\omega_2)$ is strictly weaker than $GMP$ regarding the existence of weak Kurepa trees.
4. Open Questions: The authors leave open the question of whether $TP(\omega_2)$ itself is destructible by ccc forcing. They note that their model destroys $GMP$ via a Kurepa tree, but $TP(\omega_2)$ can coexist with Kurepa trees (unlike $GMP$), leaving the status of $TP(\omega_2)$'s indestructibility unresolved.

5. Conclusion

Lambie-Hanson and Stejskalová successfully construct models demonstrating that the Guessing Model Property is not absolute under ccc forcing of size $\omega_1$. By ingeniously modifying Mitchell forcing to inject automorphisms into a Suslin tree, they create a scenario where $GMP$ holds but is vulnerable to destruction. Furthermore, they disentangle the Kurepa and Weak Kurepa hypotheses, showing that strong compactness principles can coexist with the existence of weak Kurepa trees while forbidding full Kurepa trees. These results significantly advance the understanding of the interplay between large cardinals, forcing, and combinatorial set theory at $\omega_2$.