# 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.