Ladders and Squares
Lorenzo Notaro
TL;DR
The paper advances Ditor’s ladder problems by introducing (n, κ)-semiladders and (n, κ)-ladders, alongside special ladders built via Todorčević-style ρ-functions. It proves existence results under square principles for singular cardinals and, with V=L, existence of ladders, thereby giving consistent positive answers to Ditor’s questions across regimes, including a 3-ladder of size ℵ2 from □ℵ1. The core techniques rely on quasi-product constructions and carefully controlled ρ-embeddings to manage principal-ideals and breadth, enabling iterative growth from small bases to large witnesses. The results bridge set-theoretic combinatorics and lattice representation, while outlining open problems about ZFC-provability and the relative strength of various square-like principles. Open questions remain about the necessity of square principles for certain ladder sizes and the precise landscape of ZFC consistency for Ditor’s problems.
Abstract
In 1984, Ditor asked two questions: (1) For each $n\inω$ and infinite cardinal $κ$, is there a join-semilattice of breadth $n+1$ and cardinality $κ^{+n}$ whose principal ideals have cardinality $< κ$? (2) For each $n \in ω$, is there a lower-finite lattice of cardinality $\aleph_{n}$ whose elements have at most $n+1$ lower covers? We show that both questions have positive answers under the axiom of constructibility, and hence consistently with $\mathsf{ZFC}$. More specifically, we derive the positive answers from assuming that $\square_κ$ holds for enough $κ$'s.