# 2013 seminar talk: Some problems related to Borel ideals: Towers, Luzin-type families, forcing (in)destructibility, and more

Talk held by Barnabas Farkas (KGRC) at the KGRC seminar on 2013-11-07.

### Abstract

I will discuss the possible existence of (maximal) towers in Borel ideals. I will prove the following results: Theorem 1. After adding uncountable many Cohen reals, there are towers in every tall Borel P-ideals. Theorem 2 (Brendle). It is consistent that there are no towers in any Borel P-ideals.

Related to Theorem 2, I will show that although we cannot expect more, namely "domination" from $\sigma$-centered forcing notions, the Localization forcing (which is $\sigma$-$n$-linked for every $n$) dominates every Borel P-ideals. Here the reverse implication is still an open problem.

Theorem 2 will lead us to the next natural question, namely the possible existence of so-called idealized Luzin-type families of size $\omega_2$. To obtain such a family (very probably) we need some kind of iterated destruction of the ideal without Cohen reals (at least). I will show that the Random forcing works for the summable ideal but not for the density zero ideal.