Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On some $\Sigma^{B}_{0}$-generalizations of the pigeonhole and the
  modular counting principles over $V^{0}$   Ajtai's discovery of $V^{0}\not \vdash ontoPHP^{n+1}_{n}$, where
$ontoPHP^{n+1}_{n}$ is a $\Sigma^{B}_{0}$ formalization of the statement "there
does not exist a bijection between $(n+1)$ pigeons and $n$ holes," was a
significant breakthrough in proof complexity, and there have been many
interesting generalizations and variations of this result.
  In this paper, we first focus on the following result: for any $p \geq 2$,
$V^{0}+Count^{p}_{k} \not \vdash injPHP^{n+1}_{n}$, where $Count^{p}_{k}$
denotes a $\Sigma^{B}_{0}$ formalization of the modular counting principle mod
$p$ and $injPHP^{n+1}_{n}$ denotes that of the pigeonhole principle for
injections.
  We try to make this result uniform for $p$.
  We give three types of (first-order and propositional) formulae which at
first glance seem to be generalized versions of counting principles. In
particular, we see two of them, $UCP^{l,d}_{n}$ and $GCP$, actually serve as
uniform versions of $Count^{p}_{n}$ ($p \geq 2$).
  Then we conjecture that $V^{0}+UCP^{l,d}_{k} \not \vdash injPHP^{n+1}_{n}$,
and give a sufficient condition to prove it.
  We also compare the strengths of oddtown theorem and Fisher's inequality with
those of the modular counting principles over $V^0$ by stating some conjectures
and giving sufficient conditions to prove them.

31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
15
The categories of this paper are: nucl-ex