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 $\operatorname{GPHP}(\Sigma_{n+1})$ (the $\Sigma_{n+1}$-instance of Kaye's General Pigeonhole Principle) lies strictly between $\operatorname{CARD}(\Sigma_{n+1})$ and $\operatorname{WPHP}(\Sigma_{n+1})$ (Weak Pigeonhole Principle for $\Sigma_{n+1}$-maps), and also that $\operatorname{FRT}(\Sigma_{n+1})$ (Finite Ramsey's Theorem for $\Sigma_{n+1}$-maps) does not imply $\operatorname{WPHP}(\Sigma_{n+1})$.