### Order/Disorder - Ramsey Theory

Posted:

**Thu Apr 06, 2017 2:34 pm**Dear Colleagues,

While it might seem that obtaining important results in combinatorics might have come before the spectacular development of Calculus and subsequent work in Real and Complex Analysis, discrete mathematics to which combinatorics in large measure belongs, developed later than the "continuous" mathematics needed for Calculus.

Sometimes Ramsey Theory (named for Frank Ramsey) is described by saying that mathematics does not allow "total disorder." If one two colors the edges of a complete graph on 6 vertices (red edges indicated people who are acquaintances, and blue edge those who don't know each other), then one can either find three people (who the vertices represent) who are mutual acquaintances or mutual non-acquaintances. This is expressed by saying that R(3,3)= 6. R(4,4) = 18 and amazingly R(5,5) is still not determined but here is an expository paper that mentions recent progress on this problem:

https://anthonybonato.com/2017/04/05/br ... ey-theory/

The technical paper is here:

https://arxiv.org/abs/1703.08768

Regards,

Joe

While it might seem that obtaining important results in combinatorics might have come before the spectacular development of Calculus and subsequent work in Real and Complex Analysis, discrete mathematics to which combinatorics in large measure belongs, developed later than the "continuous" mathematics needed for Calculus.

Sometimes Ramsey Theory (named for Frank Ramsey) is described by saying that mathematics does not allow "total disorder." If one two colors the edges of a complete graph on 6 vertices (red edges indicated people who are acquaintances, and blue edge those who don't know each other), then one can either find three people (who the vertices represent) who are mutual acquaintances or mutual non-acquaintances. This is expressed by saying that R(3,3)= 6. R(4,4) = 18 and amazingly R(5,5) is still not determined but here is an expository paper that mentions recent progress on this problem:

https://anthonybonato.com/2017/04/05/br ... ey-theory/

The technical paper is here:

https://arxiv.org/abs/1703.08768

Regards,

Joe