Summary: Combinatorics in Bounded Arithmetic
June 23 2004
Jeremy Avigad (Advisor)
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: com-
binatorics. 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