It is well-known that the statement "all $\aleph_1$-Aronszajn trees are special'' is consistent with ZFC (Baumgartner, Malitz, and Reinhardt), and even with ZFC+GCH (Jensen). In contrast, Ben-David and Shelah proved that, assuming GCH, for every singular cardinal $\lambda$: if there exists a $\lambda^+$-Aronszajn tree, then there exists a non-special one. Furthermore:
Theorem (Ben-David and Shelah, 1986) Assume GCH and that $\lambda$ is singular cardinal. If there exists a special $\lambda^+$-Aronszajn tree, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
This suggests that following stronger statement:
Conjecture. Assume GCH and that $\lambda$ is singular cardinal.
If there exists a $\lambda^+$-Aronszajn tree,
then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.

The assumption that there exists a $\lambda^+$-Aronszajn tree is a very mild square-like hypothesis (that is, $\square(\lambda^+,\lambda)$).
In order to bloom a $\lambda$-distributive tree from it, there is a need for a toolbox, each tool taking an abstract square-like sequence and producing a sequence which is slightly better than the original one.
For this, we introduce the monoid of postprocessing functions and study how it acts on the class of abstract square sequences.
We establish that, assuming GCH, the monoid contains some very powerful functions. We also prove that the monoid is closed under various mixing operations.
This allows us to prove a theorem which is just one step away from verifying the conjecture:

Theorem 1. Assume GCH and that $\lambda$ is a singular cardinal.
If $\square(\lambda^+,<\lambda)$ holds, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
Another proof, involving a 5-steps chain of applications of postprocessing functions, is of the following theorem.

Theorem 2. Assume GCH. If $\lambda$ is a singular cardinal and $\square(\lambda^+)$ holds, then there exists a $\lambda^+$-Souslin tree which is coherent mod finite.

This is joint work with Ari Brodsky. See: http://assafrinot.com/paper/29[-]

Kiesler and Tarski characterized weakly compact cardinals as those inaccessible cardinals such that for every $\kappa$-complete subalgebra $\mathcal{B}\subseteq P(\kappa))$ every $\kappa$-complete filter on $\mathcal{B}$ can be extended to a $\kappa$-complete ultrafilter on $\mathcal{B}.$ Welch proposed a variant of Holy-Schlict games where, for a fixed $\gamma$, player I and II take turns, with I playing an increasing sequence of subalgebras $\mathcal{A}_{\mathrm{i}}$ and II playing an increasing sequence of ultrafilters $\mathcal{U}_{\mathrm{i}}$ for $ i<\gamma$. Player II wins if she can continue playing of length $\gamma.$
By Kiesler-Tarski, player II wins the game with $\gamma=\omega$ if and only if $\kappa$ is weakly compact. It is immediate that if $\kappa$ is measurable, then II wins the game of length $2^{\kappa}$. Are these the only cases?
Nielsen and Welch proved that if II has a winning strategy in the game of length $\omega+1$ then there is an inner model with a measurable cardinal. Welch conjectured that if II has a winning strategy in the game of length $\omega+1$ then there is a precipitous ideal on $\kappa$ .
Our first result confirms Welch's conjecture: if II has a winning strategy in the game of length $\omega+1$ then there is a normal, $\kappa$-complete precipitous ideal on $\kappa$ . In fact if $\gamma\leq\kappa$ is regular and II wins the game of length $\gamma$, then there is a normal, $\kappa$-complete ideal on $\kappa$ with a dense tree that is $<-\gamma$-closed.
But is this result vacuous? Our second result is that if you start with a model with sufficient fine structure and a measurable cardinal then there is a forcing extension where:
1. $\kappa$ is inaccessible and there is no $\kappa^{+}$-saturated ideal on $\kappa$,
2. for each regular $\gamma\leq\kappa$, player II has a winning strategy in the game of length $\gamma,$
3. for all regular $\gamma\leq\kappa$ there is a normal fine ideal $\mathcal{I}_{\gamma}$ such that $P(\kappa)/\mathcal{I}\gamma$ has a dense, $<-\gamma$ closed tree.
The proofs of these results use techniques from the proofs of determinacy, lottery forcing, iterated club shooting and new techniques in inner model theory. They leave many problems open and not guaranteed to be difficult.
This is joint work of M Foreman, M. Magidor and M. Zeman.[-]