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

We formalize various counting principles and compare their strengths over \(V^{0}\) $V^0$ . In particular, we conjecture the following mutual independence between:

a uniform version of modular counting principles and the pigeonhole principle for injections,

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,

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.