Models of Bounded Arithmetic and variants of Pigeonhole Principle

We give elementary proof that theory $T_2^1(R)$ augmented by the weak pigeonhole principle for all ${\mathrm{\Delta }}_1^b(R)$-definable relations does not prove the bijective pigeonhole principle for $R$. This can be derived from known more general results but our proof yields a model of $T_2^1(R)$ in which $ontoPH{P}_n^{n+1}(R)$ fails for some nonstandard element $n$ while $PH{P}_m^{m+1}$ holds for all ${\mathrm{\Delta }}_1^b(R)$-definable relations and all $m\le n^{1-\epsilon }$, where $\epsilon >0$ is a fixed standard rational parameter. This can be seen as a step towards solving an open question posed by M. Ajtai.