On some $\Sigma^{B}_{0}$-formulae generalizing counting principles over $V^{0}$

We formalize various counting principles and compare their strengths over $V^{0}$. In particular, we conjecture the following mutual independence between: (1) a uniform version of modular counting principles and the pigeonhole principle for injections, (2) a version of the oddtown theorem and modular counting principles of modulus $p$, where $p$ is any natural number which is not a power of $2$, (3) and a version of Fisher's inequality and modular counting principles. Then, we give sufficient conditions to prove them. We give a variation of the notion of $PHP$-tree and $k$-evaluation to show that any Frege proof of the pigeonhole principle for injections admitting the uniform counting principle as an axiom scheme cannot have $o(n)$-evaluations. As for the remaining two, we utilize well-known notions of $p$-tree and $k$-evaluation and reduce the problems to the existence of certain families of polynomials witnessing violations of the corresponding combinatorial principles with low-degree Nullstellensatz proofs from the violation of the modular counting principle in concern.