
## 2. Ultrafilters

1. #### Problem 2.1.

[David Ross] Does the choice of specific ultrafilter have an affect on the resulting model of nonstandard analysis? Can special properties of ultrafilters (e.g. idempotence, $P$-points, selective) be utilized for proving combinatorial results?
•     An element $\alpha\in{^*}\mathbb{N}$ is idempotent if, for any $A\subseteq\mathbb{N}$, whenever $\alpha\in{^*}A$ then $\alpha+a\in{^*}A$ for some $a\in A$.

#### Problem 2.2.

[Mauro Di Nasso] Give a model theoretic proof of the existence of idempotent points in ${^*}\mathbb{N}$.
•     An ultrafilter $\mathcal{U}$ on $\mathbb{N}$ is Hausdorff if for any $f,g\colon \mathbb{N}\longrightarrow\mathbb{N}$, if $f(\mathcal{U})=g(\mathcal{U})$ then $\{n\in\mathbb{N}:f(n)=g(n)\}\in\mathcal{U}$.

Assuming the Continuum Hypothesis, Hausdorff ultrafilters on $\mathbb{N}$ exist (see, e.g., [MR2207497]).

#### Problem 2.3.

[Mauro Di Nasso] Give a ZFC proof of the existence of a Hausdorff ultrafilter.

Cite this as: AimPL: Nonstandard methods in combinatorial number theory, available at http://aimpl.org/nscombinatorial.