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.

24 pages