Finite Combinatorics and Fragments of Arithmetic

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 $\mathrm{WPHP}(\Sigma_{n+1})$.