# Combinatorial principles equivalent to weak induction

## TL;DR

This paper explores the logical strength of two combinatorial principles, ERT and ECT, showing their equivalences to certain induction schemes and analyzing their computational complexity via Weihrauch degrees.

## Contribution

It provides new proofs of ERT within RCA_0, establishes their equivalences to specific induction schemes in RCA_0*, and conducts a Weihrauch degree analysis of these principles.

## Key findings

- ERT is equivalent to Σ₁ induction in RCA₀*
- ECT is equivalent to Σ₂ induction in RCA₀*
- Weihrauch analysis shows ERT and ECT correspond to specific computational degrees

## Abstract

We consider two combinatorial principles, ${\sf{ERT}}$ and ${\sf{ECT}}$. Both are easily proved in ${\sf{RCA}}_0$ plus ${\Sigma^0_2}$ induction. We give two proofs of ${\sf{ERT}}$ in ${\sf{RCA}}_0$, using different methods to eliminate the use of ${\Sigma^0_2}$ induction. Working in the weakened base system ${\sf{RCA}}_0^*$, we prove that ${\sf{ERT}}$ is equivalent to ${\Sigma^0_1}$ induction and ${\sf{ECT}}$ is equivalent to ${\Sigma^0_2}$ induction. We conclude with a Weihrauch analysis of the principles, showing ${\sf{ERT}} {\equiv_{\rm W}} {\sf{LPO}}^* {<_{\rm W}}{{\sf{TC}}_{\mathbb N}}^* {\equiv_{\rm W}} {\sf{ECT}}$.