Lecture series on Infinite Combinatorics and Combinatorial Number Theory

This series (of 5 lectures) is given by Thierry Coquand, Peter Hegarty and Jeff Steif.

At this time, it seems that the schedule will be as follows.

Week 10
Thursday, March 11, 10:15-12:00 in MD3

Week 11
Thursday, March 18, 10:15-12:00 in MD1

Week 12
Tuesday, March 23, 10:15-12:00 in S1

Week 13
Thursday, April 1, 10:15-12:00 in S1

Week 15
Thursday, April 15, 10:15-12:00 in S1

The topics are the following (in the order of whom will speak).

1. (JS)
We will show how topological dynamics and ergodic theory can
be used to give proofs of van der Waerden's theorem and Szemeredi's
theorem concerning the existence of arbitrarily long arithmetric
progressions in certain subsets of the integers. Extensions also discussed.

2. (PH)
We will discuss some problems of a combinatorial flavor in the field of
additive number theory. A Sidon set is a subset of the positive integers
for which all sums of pairs of elements are distinct. The main question is
to determine the maximum asymptotic density of such a set ; this problems
and some generalisations have seen a lot of work but are still unsolved in
many cases. If time permits, I will discuss the multiplicative analogue of
the problem which was solved completely by Erdös using graph theory.

3. (TC)
The set of all ultrafilters over N can be given a structure of
compact semigroup. Though quite non effective (the space is non separable,
and the semigroup operation non continuous in one variable), this
structure has been used to give elegant proofs of some combinatorial
statements. We present some of them, in particular a proof of Ramsey's
theorem and of an additive version of it, due to Hindman.