On collection schemes and Gaifman's splitting theorem

We study model theoretic characterizations of various collection schemes over ${\mathbf{P}\mathbf{A}}^{-}$ from the viewpoint of Gaifman’s splitting theorem. Among other things, we prove that for any $n\ge 0$ and $M\models {\mathbf{P}\mathbf{A}}^{-}$, the following are equivalent: 1. $M$ satisfies the collection scheme for ${\mathrm{\Sigma }}_{n+1}$ formulas. 2. For any $K,N\models {\mathbf{P}\mathbf{A}}^{-}$, if $M{\subseteq }_{\mathrm{c}\mathrm{o}\mathrm{f}}K$, $M{\prec }_{{\mathrm{\Delta }}_0}K$ and $M\prec N$, then $M{\prec }_{{\mathrm{\Sigma }}_{n+2}}K$ and ${\sup }_N(M){\prec }_{{\mathrm{\Sigma }}_n}N$. 3. For any $N\models {\mathbf{P}\mathbf{A}}^{-}$, if $M\prec N$, then $M{\prec }_{{\mathrm{\Sigma }}_{n+2}}{\sup }_N(M){\prec }_{{\mathrm{\Sigma }}_n}N$. Here, ${\sup }_N(M)$ is the unique $K$ satisfying $M{\subseteq }_{\mathrm{c}\mathrm{o}\mathrm{f}}K{\subseteq }_{\mathrm{e}\mathrm{n}\mathrm{d}}N$. We also investigate strong collection schemes and parameter-free collection schemes from the similar perspective.