On some Σ B₀-formulae generalizing counting principles over V⁰

We formalize various counting principles and compare their strengths over V⁰ 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.