This monograph covers some of the most important developments in Ramsey
theory from its beginnings in the early 20th century via its many
breakthroughs to recent important developments in the early 21st
century.
The book first presents a detailed discussion of the roots of Ramsey
theory before offering a thorough discussion of the role of parameter
sets. It presents several examples of structures that can be interpreted
in terms of parameter sets and features the most fundamental Ramsey-type
results for parameter sets: Hales-Jewett's theorem and
Graham-Rothschild¹s Ramsey theorem as well as their canonical versions
and several applications. Next, the book steps back to the most basic
structure, to sets. It reviews classic results as well as recent
progress on Ramsey numbers and the asymptotic behavior of classical
Ramsey functions. In addition, it presents product versions of Ramsey's
theorem, a combinatorial proof of the incompleteness of Peano
arithmetic, provides a digression to discrepancy theory and examines
extensions of Ramsey's theorem to larger cardinals. The next part of the
book features an in-depth treatment of the Ramsey problem for graphs and
hypergraphs. It gives an account on the existence of sparse and
restricted Ramsey theorem's using sophisticated constructions as well as
probabilistic methods. Among others it contains a proof of the induced
Graham-Rothschild theorem and the random Ramsey theorem. The book closes
with a chapter on one of the recent highlights of Ramsey theory: a
combinatorial proof of the density Hales-Jewett theorem.
This book provides graduate students as well as advanced researchers
with a solid introduction and reference to the field.
