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# Preprints

` `**
GUESSING MODELS AND THE APPROACHABILITY IDEAL**,

with R. Mohammadpour,

Abstract

Starting with two supercompact cardinals we produce a generic extension of the universe in which the principles $${\rm ISP}(\omega_2)$$ and $${\rm ISP}(\omega_3)$$ hold simultaneously, and the restriction of the approachability ideal $$I[\omega_2]$$ to the set of ordinals of cofinality $$\omega_1$$ is the non stationary ideal on this set.

` `**
RANKS OF MAHARAM ALGEBRAS**,

with Z. Perovic,

Abstract

Solving a well-known problem of Maharam, Talagrand constructed an
exhaustive non uniformly exhaustive submeasure, thus also providing the first example
of a Maharam algebra that is not a measure algebra. To each exhaustive submeasure
one can canonically assign a certain countable ordinal, its exhaustivity rank. In this
paper, we use carefully constructed Schreier families and norms derived from them to
provide examples of exhaustive submeasures of arbitrary high exhaustivity rank. This
gives rise to uncountably many non isomorphic separable atomless Maharam algebras.

` `**
ITERATION OF SEMIPROPER FORCING REVISITED**,

Abstract

We present a method for iterating semiproper forcing
which uses side conditions and is inspired by the technique recently
introduced by Neeman.

```
```**PROPER FORCING REMASTERED**,

with G. Venturi,

Abstract

In these notes we present the method introduced by Neeman of generalized side conditions with two types of models.
We then discuss some applications: the Friedman-Mitchell poset for adding a club with finite conditions, Koszmider's forcing
construction of a strong chain of functions from \omega_1 to \omega_1, and the Baumgartner-Shelah forcing construction of
a thin very tall superatomic Boolean algebra.

```
```**
OSCILLATIONS AND THEIR APPLICATIONS IN PARTIION CALCULUS**,

with L. Fontanella,

Abstract

Lectures notes taken by L. Fontanella from a tutorial given
at the 2nd Young Set Theory Workshop,
April 14-18, 2009 at the CRM in Bellaterra, Spain.

# Notes

```
```**Jeux infinis et détermination**,

Abstract

Lecture notes from a course on Infinite games and determinacy tought at the Université de Paris - Diderot, 2009/2010 (in french).

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