Finite Combinatorics and Fragments of Arithmetic

TL;DR

This paper explores the combinatorial properties of definable maps in fragments of first-order arithmetic, revealing their relationships and independence results concerning pigeonhole principles and finite Ramsey's theorem.

Contribution

It establishes the precise logical strength and relationships among combinatorial principles like GPHP, CARD, WPHP, and FRT within certain arithmetic fragments.

Findings

GPHP(Σ_{n+1}) is strictly between CARD(Σ_{n+1}) and WPHP(Σ_{n+1})

FRT(Σ_{n+1}) does not imply WPHP(Σ_{n+1})

Definable maps behave differently in finite domains without BΣ_{n+1}

Abstract

In fragments of first order arithmetic, definable maps on finite domains could behave very differently from finite maps. Here combinatorial properties of $\Sigma_{n+1}$-definable maps on finite domains are compared in the absence of $B\Sigma_{n+1}$. It is shown that $\mathrm{GPHP}(\Sigma_{n+1})$ (the $\Sigma_{n+1}$-instance of Kaye's General Pigeonhole Principle) lies strictly between $\mathrm{CARD}(\Sigma_{n+1})$ and $\mathrm{WPHP}(\Sigma_{n+1})$ (Weak Pigeonhole Principle for $\Sigma_{n+1}$-maps), and also that $\mathrm{FRT}(\Sigma_{n+1})$ (Finite Ramsey's Theorem for $\Sigma_{n+1}$-maps) does not imply $\mathrm{WPHP}(\Sigma_{n+1})$.