Bounded Arithmetic (Buss' $S_2$ and its subsystems) is significant for at least two reasons:

1. Its close connection to computational complexity, and
2. The fact that the exponentiation function cannot be defined.

The latter reason is significant because many of the theorems of combinatorics and basic number theory can be proven in a straightforward manner if we add an axiom to $S_2$ stating that the exponentiation function is total. So it is an interesting question to consider what theorems can be proven in just $S_2$ , without exponentiation. I have focused on Ramsey theory and the probabilistic methods, in the context o f$S_2$ and its subsystems.

In order to deal with probabilistic methods in the absence of exponentiation, I will use the weak pigeonhole principle to simulate the counting argument. I use the fact that we can mimic the counting argument by a bounded formula. In effect, the non-constructive counting argument, has some constructive content. Many of these theorems have some bound in the statement of the theorem. In order to prove the theorems in $S_2$, the bound must often be made less tight; furthermore, different subsystems of $S_2$ can prove different bounds. An interesting phenomenon arises: A trade-off between the strength of the bound in a theorem versus the strength of the subsystem used to prove the theorem.

To this point, I have considered what can be proven in $S_2$ and its subsystems. We would of course also want to know what cannot be proven. In some cases such independence results would solve the hard open question of the non-collapse of the subsystems of $S_2$. I consider a seemingly more feasible alternative: Finding reversals. Often the proof of a theorem will rely heavily on some principle. Borrowing from the terminology of Reverse Mathematics, I use the term reversal to refer to the claim that the theorem in fact implies this principle used to prove it. I will discuss two reversals.