Reverse mathematics and dimension of posets

TL;DR

This paper explores the reverse mathematics of order dimension bounds in posets, establishing equivalences and provability results within subsystems of second-order arithmetic.

Contribution

It introduces and analyzes principles bounding poset dimension via subposets, connecting them to well-known subsystems like WKL_0 and IΣ^0_2.

Findings

Both $ ext{DBi}_n$ and $ ext{DBc}_n$ are equivalent to WKL_0.

$ ext{DB}_p$ and $ ext{DB}^+_p$ are provable from WKL_0 and IΣ^0_2.

$ ext{B} ext{Sigma}^0_2$ does not prove $ ext{DB}^+_p$.

Abstract

Order dimension theory measures the complexity of partially ordered sets by quantifying how far they are from being linearly ordered. In this paper we study classical bounding results for order dimension within the framework of reverse mathematics. We focus on principles asserting that the dimension of a poset can be bounded in terms of the dimension of subposets obtained by removing chains or points, denoted by $\mathsf{DBi_n}$, $\mathsf{DBc_n}$, and $\mathsf{DB_p}$. We prove that, over $\mathsf{RCA}_0$, both $\mathsf{DBi_n}$ and $\mathsf{DBc_n}$ are equivalent to $\mathsf{WKL}_0$. To analyze $\mathsf{DB_p}$, we introduce a natural strengthening $\mathsf{DB^+_p}$ and show that both $\mathsf{DB_p}$ and $\mathsf{DB^+_p}$ are provable from $\mathsf{WKL}_0$ and from $\mathsf{I}\Sigma^0_2$, while $\mathsf{B}\Sigma^0_2$ does not suffice to prove $\mathsf{DB^+_p}$. The latter result is obtained by showing that the statement \lq\lq $\mathsf{DB^+_p}$ is computably true\rq\rq\ is equivalent to $\mathsf{I}\Sigma^0_2$.