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In this paper we prove $T_{2}^{1}(R) + WPHP(\Delta_{1}^{p}(R)) \nvdashontoPHP(R)$ by means of forcing, where $ontoPHP(R)$ denotes the bijectivepigeonhole principle formulated for a binary relation symbol $R$ and$WPHP(\Delta_{1}^{p}(R))$ denotes the weak pigeonhole principle stated for allpolynomial-time decidable relations with oracle access to $R$. As a by-productwe construct a model which satisfies: \begin{align*}T_{2}^{1}(R) + \forall m\leq n^{1 - \epsilon} (PHP_{m}^{m + 1}(\Delta_{1}^{p}(R))) + \neg PHP_{n}^{n +1}(R),\end{align*} where $n$ is non-standard and $\epsilon$ is arbitrarilysmall (but standard). This can be seen as a step towards a solution of an openquestion posed by M. Ajtai.

Models of Bounded Arithmetic and variants of Pigeonhole Principle