Finite Combinatorics and Fragments of Arithmetic

In fragments of first order arithmetic, definable maps on finite domains could behave very differently from finite maps. Here combinatorial properties of $Σ_{n+1}$-definable maps on finite domains are compared in the absence of $BΣ_{n+1}$. It is shown that $\mathrm{GPHP}(Σ_{n+1})$ (the $Σ_{n+1}$-ins