Ramsey theory refers to a branch of mathematics whose underlying philosophy is captured by the statement that "Every large system contains a large well-organized subsystem." There are examples of such statements in many areas, including geometry, number theory and analysis.
In this talk, we shall discuss some examples and focus on Roth's theorem, which guarantees that every set of integers with positive density contains a 3-term arithmetic progression. No prior knowledge of combinatorics and number theory is assumed.