#paper #square #tree #souslin #aronszajn # Abstract Abstract. Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^{\lambda}=\lambda ^{+}$ , then $\square ^{*}_{\lambda}$ entails the existence of a normal $\lambda$-distributive $\lambda ^{+}$-Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis $\square ^{*}_{\lambda}=\square_{\lambda}(\lambda^{+},<\lambda^{+})$ by $\square(\lambda^{+},<\lambda)$. As $\square(\lambda^{+},<\lambda)$ does not impose a bound on the order-type of the witnessing clubs, our construction is necessarily different from that of Ben-David and Shelah, and instead uses walks on ordinals augmented with club guessing. A major component of this work is the study of postprocessing functions and their effect on square sequences. A byproduct of this study is the finding that for 𝜅 regular uncountable, $\square(\kappa)$ entails the existence of a partition of 𝜅 into 𝜅 many fat sets. When contrasted with a classical model of Magidor, this shows that it is equiconsistent with the existence of a weakly compact cardinal that $𝜔_{2}$ cannot be split into two fat sets. # 1. Partitioning a fat set Introduction to $\mathcal{C}$-sequences and postprocessing functions. Proof of: **Theorem.** If $\square(\kappa)$ holds, then every fat subset of 𝜅 may be partitioned into 𝜅 many fat sets. In particular, the failure to partition $𝜔_2$ into two fat sets is equiconsistent with the existence of a weakly compact cardinal. # 2. **Theorem.** Assume GCH, and that $𝜆$ is some singular cardinal. If $\square(\lambda^{+},<\lambda)$ holds, then there exists a normal $𝜆$-distributive $𝜆^+$-Aronszajn tree. # 3. **Theorem.** Assume GCH, and that $𝜆$ is some singular cardinal. If $\square_{\lambda}(\lambda^{+},<\lambda^{+})$ holds, then there exists a normal $𝜆$-distributive $𝜆^+$-Aronszajn tree. # 4. **Theorem.** Suppose that 𝜆 is a strong-limit singular cardinal and $2^{\lambda}=\lambda ^{+}$. If $\square(\lambda^{+},<2)$ holds, then there exists a uniformly coherent $𝜆^+$-Souslin tree.