# Tree property using side conditions

The following problems were asked during the high and low forcing workshop:

Question 1. Can one force tree property at $\kappa^{++}$ for $\kappa$ singular using side conditions?

Question 2. Can one force tree property at successive cardinals using side conditions?

As it is stated in A report on the workshop activities, question 1 is solved during the conference.

I wonder to know if there is any reference for the proof, or if someone can explain the basic ideas of the proof.

I am also interested to know if any progress on question 2 is obtained? (I may mention that in Nemman's paper Forcing with Sequences of Models of Two Types, this is claimed for two successive cardinals but no details are given).

• I have two terminological qualms here: (1) it wasn't a conference it was a workshop; and (2) by side conditions do you mean Itay's constructions and the related ideas, or just the general term for a forcing with some additional structural conditions? (E.g. coding a subset of $\omega_1$ into a real is a Cohen forcing with side conditions.) – Asaf Karagila Jul 17 '16 at 17:27
• (Also I recognize at least a handful of faces which have active MO accounts, maybe one of them will pass around and answer. Which I'm sure is what you were hoping for, more or less.) – Asaf Karagila Jul 17 '16 at 17:29
• @AsafKaragila Thanks, I changed conference to workshop. By side condition I mean Neeman's method. – Mohammad Golshani Jul 18 '16 at 4:19
• In Boban Velickovic's talk at the Magidor 70 conference, he mentions (at the very end) that by an application of his pure side condition poset, he can get a model like Uri Abraham's, with the tree property holding at both $\aleph_2$ and $\aleph_3$. I don't think Neeman mentions this application in his original paper, only how to get the tree property at a single cardinal. – tci Jul 19 '16 at 1:49