# Ramsey theory

Tuesday, September 30, 2014 - 2:00pm - 2:50pm

Vitaly Bergelson (The Ohio State University)

By analogy with the classical notions of density in the set N of natural numbers, one can introduce notions of density which are geared towards the multiplicative structure of N. Various combinatorial results involving additively large sets in (N,+) ( such as, for instance, Szemeredi's theorem on arithmetic progressions and its polynomial extensions) have natural analogs in the multiplicative semigroup (N,x). For example, muItiplicatively large sets in N contain arbitrarily long geometric progressions.

Thursday, September 11, 2014 - 3:15pm - 4:05pm

David Conlon (University of Oxford)

Ramsey's theorem states that for any graph H there exists an n such that any 2-colouring of the complete graph on n vertices contains a monochromatic copy of H. Moreover, for large n, one can guarantee that there are at least c_H n^v copies of H, where v is the number of vertices in H. In this talk, we investigate the problem of optimising the constant c_H, focusing in particular on the case of complete graphs.

Saturday, December 1, 2012 - 9:00am - 10:00am

János Pach (Hungarian Academy of Sciences (MTA))

By Ramsey's theorem, any system of n segments in the plane

has roughly logn members that are either pairwise disjoint or pairwise

intersecting. Analogously, any set of n points p(1),..., p(n) in the

plane has a subset of roughly loglogn elements with the property that

the orientation of p(i)p(j)p(k) is the same for all triples from this

subset with i less than j less than k. (The elements of such a subset form the vertex set

of a convex polygon.) However, in both cases we know that there exist

has roughly logn members that are either pairwise disjoint or pairwise

intersecting. Analogously, any set of n points p(1),..., p(n) in the

plane has a subset of roughly loglogn elements with the property that

the orientation of p(i)p(j)p(k) is the same for all triples from this

subset with i less than j less than k. (The elements of such a subset form the vertex set

of a convex polygon.) However, in both cases we know that there exist