  This work is concerned with an axiom introduced by Todorcevic in [9] that
constitutes a Ramsey-like statement regarding the topology of the reals. Our
aim is to explain the axiom in detail, give some interesting applications and
finally prove that the axiom is indeed consistent with ZFC, so that it makes
sense to consider working with it in the first place.
  For this particular academic endeavor, we cover several advanced topics in
set theory, including concepts like Hausdorff gaps, forcing, infinitary
combinatorics and a tad of topology. We employ, for example, an argument based
on Rothberger's theorem to show that the Open Coloring Axiom implies the
equality $\mathfrak b=\aleph_1$, which in turn makes this axiom inconsistent
with CH. In other words, in ZFC, the Open Coloring Axiom could be false. To
prove its relative consistency, we show that the axiom could be true by
following a rather long and technical lemma of Todorcevic, which leads to the
culmination of this work.
