Problems in Ramsey theory typically ask a question of the form: "how big must some structure be to guarantee that a particular property holds?
"[1] A typical result in Ramsey theory starts with some mathematical structure that is then cut into pieces.
A classic reference for these and many other results in Ramsey theory is Graham, Rothschild, Spencer and Solymosi, updated and expanded in 2015 to its first new edition in 25 years.
Secondly, while Ramsey theory results do say that sufficiently large objects must necessarily contain a given structure, often the proof of these results requires these objects to be enormously large – bounds that grow exponentially, or even as fast as the Ackermann function are not uncommon.
In other cases it is known that any bound must be extraordinarily large, sometimes even greater than any primitive recursive function; see the Paris–Harrington theorem for an example.
In other cases, the reason behind a Ramsey-type result is that the largest partition class always contains the desired substructure.