On maximal ladders
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915921388371968 |
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| author | Notaro, Lorenzo |
| author_facet | Notaro, Lorenzo |
| contents | 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 asked whether this bound is sharp, i.e., whether for each $n$ there is an $n$-ladder of cardinality $\aleph_{n-1}$. We isolate the notion of maximal $n$-ladder and use it to study Ditor's problem and related questions. We show that $\text{Add}(ω, ω_ω)$ forces every maximal $n$-ladder to have cardinality $\aleph_{n-1}$, and hence forces a positive answer to Ditor's question for every $n$. In particular, it is consistent that there are no maximal $3$-ladders of cardinality $\aleph_1$. However, we show that the existence of such a ladder follows from $\mathfrak{d}=\aleph_1$. Under $\clubsuit$, we construct a maximal $3$-ladder of breadth $2$. Finally, we prove that, consistently (under $\diamondsuit$), there exists a maximal $3$-ladder that is destructible by forcing with a Suslin tree. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_06031 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On maximal ladders Notaro, Lorenzo Combinatorics Logic 03E05 (Primary) 03E35, 06A07 (Secondary) 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 asked whether this bound is sharp, i.e., whether for each $n$ there is an $n$-ladder of cardinality $\aleph_{n-1}$. We isolate the notion of maximal $n$-ladder and use it to study Ditor's problem and related questions. We show that $\text{Add}(ω, ω_ω)$ forces every maximal $n$-ladder to have cardinality $\aleph_{n-1}$, and hence forces a positive answer to Ditor's question for every $n$. In particular, it is consistent that there are no maximal $3$-ladders of cardinality $\aleph_1$. However, we show that the existence of such a ladder follows from $\mathfrak{d}=\aleph_1$. Under $\clubsuit$, we construct a maximal $3$-ladder of breadth $2$. Finally, we prove that, consistently (under $\diamondsuit$), there exists a maximal $3$-ladder that is destructible by forcing with a Suslin tree. |
| title | On maximal ladders |
| topic | Combinatorics Logic 03E05 (Primary) 03E35, 06A07 (Secondary) |
| url | https://arxiv.org/abs/2604.06031 |