We investigate the relationship between various versions of Ramsey’s theorem and bounding schemes in a model $\mathcal {N}$ of a fragment of arithmetic $F$. Our primary objective is to recast and extend the seminal results of Hirst [15] (see Theorem 1) through the modern lens of Weihrauch reducibility. By extracting explicit Weihrauch reductions from classical proofs, we expose the uniform computational content underlying these combinatorial principles, yielding a deeper and more refined understanding of their reverse mathematical strength.

Our results, informally stated in our terminology and established inside $\mathcal {N}$, are as follows: The following are equivalent: $\mathrm {B}\Sigma ^0_2$, the statement that the finite union of finite c.e. sets is finite, and the Infinite Pigeonhole Principle (see Theorem 3). We also discuss the Weihrauch relations between these logically equivalent principles (see Section 4). The Infinite Pigeonhole Principle is Weihrauch reducible to $\mathrm {RT}^2_2$ (see Theorem 4). There is also another principle logically equivalent to $\mathrm {B}\Sigma ^0_2$ which is Weihrauch reducible to $\mathrm {SRT}^2_2$ (see Theorem 5). We show that there is a principle which is equivalent to $\mathrm {B}\Sigma ^0_3$ (see Theorem 6) and Weihrauch reducible to $\mathrm {SRT}^2_{<\infty }$ (Theorem 7). We discuss some equivalences with $\mathrm {B}\Sigma ^0_{n}$ (see Section 6.1) and end with a problem Weihrauch reducible to $\mathrm {RT}^{n+1}_{2}$ (Section 6.2).

Since we work within the model $\mathcal {N}$, many standard definitions must be adjusted. Due to the expository nature of this article, these definitions are introduced throughout the text as needed. Reading the article from start to finish will provide a better understanding of the ideas involved than focusing only on individual theorems.

This paper estimates the effects of teenage childbearing on education, working, physical and mental health, and physical activity of young girls in Mexico using two waves of the nationally representative Mexican Family Life Survey. We employ a propensity score matching model that accounts for a rich set of baseline covariates that predict teenage childbearing to attempt to reduce the bias due to confounding variables associated with teenage childbearing. The results demonstrate that teenage childbearing is associated with an increase in the probability of being overweight, and reductions in physical activity and the probability of high school completion. Moreover, the results are consistent when we employ sibling fixed effects to account for unobservable family background.