Mauro di Nasso, Università di Pisa
Nonstandard natural numbers in arithmetic Ramsey Theory and topological dynamics
The use of nonstandard models *N of the natural numbers has recently found several applications in arithmetic Ramsey theory. The basic observation is that every infinite number in *N corresponds to an ultrafilter on N, and the algebra of ultrafilters is a really powerful tool in this field. Note that this notion also makes sense in any model of PA, where one can consider the 1-type of any infinite number.
Furthermore, nonstandard natural numbers are endowed with a natural compact topology, and one can apply the methods of topological dynamics considering the shift operator $x \mapsto x+1$ . This very peculiar dynamic has interesting characteristics.
In this talk I will also present a new result in the style of Hindman’s Theorem about the existence of infinite monochromatic configurations in any finite coloring of the natural numbers. A typical example is the following monochromatic pattern:
a, b, c, $\ldots$ , a+b+ab, a+c+ac, b+c+bc, $\ldots$ , a+b+c+ab+ac+bc+abc.