What is Ramsey Theory ? what is its own importance in maths?

Question

3 days ago , i had a discussion with a close friend who studies physics - still a student - . and i was telling her about the biggest known numbers in maths , so i told her about numbers such googol and googolplex, then about Graham number and she asked me , what is the application of this number in maths problems ? i answered , it's a solution of a famous problem in Ramsey theory and Ramsey theory is a branch if modern mathematics .

but suddenly , the question arise ! what is Ramsey theory ? what is its importance ? has it any relation with other branches like mathematical logic or group theory ?

is it advanced topic in maths ?

so i have googled and found that it's a branch which talk about the properties of particular structures in maths when we divide this structure into new substructures .

this is all what i could understand !

so can give us more information , reference for beginners , application of it in other branches in science like physics or chemistry , the relation with other mathematical branches and resources for more readings ?

Answer 1

Ramsey theory has had a major impact in Banach space theory. Its first application, it seems, was in the proof of the justly famous Rosenthal $\ell_1$-theorem. Relatively recently, it was used by Gowers in his dichotomy result.

This book seems to contain the major recent results.

Joeseph Diestel's text Sequences and Series in Banach Spaces contains a nice treatment of some basic results from Ramsey Theorey and several of the applications thereof, including a proof of the Rosenthal $\ell_1$ theorem, to the geometry of Banach spaces.

This article contains many open problems of a Ramsey flavor.

Some other surveys:

1) Ramsey Methods in Banach Spaces, W.T Gowers, contained in chapter 24 of Handbook of the Geometry of Banach Spaces, Vol 2.

2) Applications of Ramsey Theorems to Banach Space Theorey, Edward Odell, 1981, Austin: University of Texas Press.

Of course, none of this is easy. Gowers won the field medal for his dichotomy result. The following quote from his biography seems relevant:

"William Timothy Gowers has provided important contributions to functional analysis, making extensive use of methods from combination theory. These two fields apparently have little to do with each other, and a significant achievement of Gowers has been to combine these fruitfully."

Answer 2

Funny that you should mention mathematical logic. Frank Plumpton Ramsey introduced (what came to be known as) Ramsey's theorem in a 1928 paper title "On a Problem of Formal Logic". Ramsey applied his theorem to solve a certain decision problem in predicate logic. If I remember right, he used it to show that the problem, whether a given first-order sentence of a certain kind has an infinite model, is effectively decidable.

Edited to add the modifier "of a certain kind" because it sure won't work for arbitrary first order sentences. I seem to recall now that Ramsey's decision procedure applies to universal (or existential-universal) first-order sentences in a purely relational language, i.e., no operation symbols. Or something like that.