A solution to Ditor's problem

Abstract: We settle the long-standing open question whether there exists a $3$-ladder of cardinality $\aleph_2$. Given a positive integer $n$, an $n$-ladder is a lower finite lattice whose elements have at most $n$ lower covers. In 1984, Ditor proved that every $n$-ladder has cardinality at most $\aleph_{n-1}$, and that this cardinal bound is sharp for $n = 1,2$. He then raised the question of whether the bound is attained for $n\ge 3$ as well. An affirmative answer is known to be consistent with $\mathsf{ZFC}$. We prove, relative to the consistency of a Mahlo cardinal, that the question is independent of $\mathsf{ZFC}$. More precisely, we show that the nonexistence of a $3$-ladder of cardinality $\aleph_2$ is equiconsistent with a Mahlo cardinal.

• The paper establishes that under a Mahlo cardinal, the existence of 3-ladders of cardinality ℵ₂ is independent of ZFC.
• It employs forcing with Coll(ω₁,<κ) alongside an analysis of join-semilattices and local breadth invariants to derive key nonexistence results.
• The work bridges combinatorial lattice theory and set theory by equating the nonexistence of such lattices with the consistency strength of Mahlo cardinals.

A Solution to Ditor’s Problem: Independence via Large Cardinals

Background and Statement of Ditor’s Problem

The study of $n$-ladders—a class of lower finite lattices in which each element has at most $n$ lower covers—captures foundational questions about the cardinalities attainable by these combinatorial objects. Ditor, in 1984, proved that every $n$-ladder has cardinality at most $\aleph_{n-1}$, and this bound is sharp for $n = 1,2$, with explicit constructions yielding $n$-ladders of cardinality $\aleph_{n-1}$ in those cases. The central question, termed Ditor’s Problem, is whether this upper bound is sharp for all $n\!>\!2$—specifically, whether there exists a $3$-ladder of cardinality $\aleph_2$.

It is known that the existence of such ladders is consistent with ZFC under certain combinatorial or large cardinal assumptions, but it has remained open whether the existence of a $n$0-ladder of cardinality $n$1 is provable in ZFC or independent of it.

Main Results and Methods

The paper settles Ditor’s Problem by showing that, relative to the consistency of a Mahlo cardinal, the existence of a $n$2-ladder of cardinality $n$3 is independent of ZFC. More precisely, the principal theorem establishes that in a model obtained by collapsing a Mahlo cardinal $n$4 to become $n$5 via $n$6, there are no lower finite lattices of breadth $n$7 and cardinality $n$8. Further, the nonexistence of such a structure is shown to be equiconsistent with the existence of a Mahlo cardinal, tightly calibrating the large cardinal strength required.

The methodology interweaves combinatorial and set-theoretic tools, including an in-depth analysis of join-semilattices, the structure of their ideals and quotients, and the behavior of filters generated from “projected upper cones”. Key invariants such as breadth and local breadth are systematically deployed, and Ditor's cardinality theorem is generalized and refined through quotient analysis. Forcing with $n$9 is analyzed via elementary submodels and chain conditions, while game-theoretic techniques (notably Laflamme’s game for meager filters) are used to characterize properties of filter bases in the collapsed models.

Sharpness and Consistency Strength

The paper demonstrates that the large cardinal hypothesis is optimal: the existence of a $n$0-ladder of cardinality $n$1 follows from $n$2, so in any model lacking such a ladder, $n$3 must fail, which in turn implies that $n$4 is Mahlo in $n$5. Hence, the consistency strength of the nonexistence result aligns precisely with the existence of a Mahlo cardinal.

Equivalence of Varied Formulations

A robust equivalence result is proven: the following are equiconsistent—

1. ZFC + "there is a Mahlo cardinal"
2. ZFC + "there are no lower finite lattices of breadth 3 and cardinality $n$6"
3. ZFC + "there are no $n$7-ladders of cardinality $n$8"

This tightens the link between structural combinatorics of lattices and high-level set-theoretic axioms.

Technical Innovations

Local Breadth and Quotient Structure

The concept of local breadth at a point $n$9 in a join-semilattice is introduced and shown to govern the propagation of upper cardinality bounds to quotients by ideals. The author proves that, for every lower finite join-semilattice of maximum possible cardinality and given ideal, one can find an element such that its join with the ideal is a chain, indicating a structural sparseness forced by optimal cardinal bounds.

Filters, Projected Upper Cones, and Rudin-Blass Reducibility

A central device is the study of filters $\aleph_{n-1}$0 generated by projected upper cones relative to countable ideals $\aleph_{n-1}$1. The paper proves that the non-meagerness (non-Baire category) of these filters is invariant among ideals via Rudin-Blass reducibility, and, in favorable situations, these filters are $\aleph_{n-1}$2-filters. This analysis is essential in the forcing argument excluding the possibility of large $\aleph_{n-1}$3-ladders in the collapse model.

Forcing and Games

The decisive technical step uses a forcing argument, combined with Talagrand’s and Laflamme’s game-theoretic characterizations of meager filters, to transfer combinatorial obstructions into models where Mahlo cardinals have been collapsed to $\aleph_{n-1}$4. Through a construction involving dense subsets and elementary submodels, the author follows the possible trajectories of cofinal and join-semilattice-generated subsets under forcing, establishing key nonexistence results via contradictions derived from filter properties.

Implications and Open Problems

The independence of the existence of $\aleph_{n-1}$5-ladders of cardinality $\aleph_{n-1}$6 has consequences both for combinatorial lattice theory and for set theory, highlighting the delicate granularity at which the structure of infinite lattices interacts with the global axioms of set theory. The consistency threshold associated with Mahlo cardinals marks a boundary for what can be achieved combinatorially at uncountable cardinalities within the framework of ZFC.

Practically, applications of $\aleph_{n-1}$7-ladders for $\aleph_{n-1}$8 in universal algebra and beyond remain elusive, and the fine structure theory provided here may illuminate or preclude possible representation theorems involving lattices of high cardinality or breadth.

Theoretically, new conjectures and questions are raised: for instance, whether $\aleph_{n-1}$9 implies the existence of a $n = 1,2$0-ladder of cardinality $n = 1,2$1, whether any model can exclude lower finite lattices of finite breadth and cardinality $n = 1,2$2 altogether, and the link between the existence of a $n = 1,2$3-ladder and the mere existence of a lower finite lattice of breadth $n = 1,2$4 of the same cardinality.

Further, questions touching on maximal $n = 1,2$5-ladders and their indestructibility under $n = 1,2$6-closed forcing, as well as the analogues of Ditor’s Problem at singular cardinals (e.g., $n = 1,2$7), are highlighted as avenues for future research, where even stronger large cardinal axioms are implicated.

Conclusion

This work provides a rigorous, fine-grained resolution of Ditor’s Problem for $n = 1,2$8 by demonstrating its independence from ZFC, contingent precisely on the existence of a Mahlo cardinal. The interplay of lattice theory, combinatorial invariants, set-theoretic forcing, large cardinals, and descriptive set theory fortifies the connection between algebraic structure and foundational axioms. The paper presents both technical generalizations—such as the local analysis of breadth and filter structure—and a clear demarcation of the limits of provability for combinatorial lattice constructions, paving the way for new explorations at the interface of algebra and set theory.

Reference: "A solution to Ditor's problem" (2606.28844)