Combinatorics in bounded arithmetic

A basic aim of logic is to consider what axioms are used in proving various theorems of mathematics. This thesis will be concerned with such issues applied to a particular area of mathematics: combinatorics. We will consider two widely known groups of proof methods in combinatorics, namely, probabilistic methods and methods using linear algebra. We will consider certain applications of such methods, both of which are significant to Ramsey theory. The systems we choose to work in are various theories of bounded arithmetic.; For the probabilistic method, the key point is that we use the weak pigeonhole principle to simulate the probabilistic reasoning. We formalize various applications of the ordinary probabilistic method and linearity of expectations, making partial progress on the Local Lemma. In the case of linearity of expectations, we show how to eliminate the weak pigeonhole principle by simulating the derandomization technique of "conditional probabilities."; We consider linear algebra methods applied to various set system theorems. We formalize some theorems using a linear algebra principle as an extra axiom. We also show how weaker results can be attained by giving alternative proofs that avoid linear algebra, and thus also avoid the extra axiom.; We formalize upper and lower Ramsey bounds. For the lower bounds, both the probabilistic methods and the linear algebra methods are used. We provide a stratification of the various Ramsey lower bounds, showing that stronger bounds can be proved in stronger theories.; A natural question is whether or not the axioms used are necessary. We provide "reversals" in a few cases, showing that the principle used to prove the theorem is in fact a consequence of the theorem (over some base theory). Thus this work can be seen as a (humble) beginning in the direction of developing the Reverse Mathematics of finite combinatorics.