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For a regular cardinal $\kappa$, a $\kappa$ tree $T$ is called special when there is a regressive function $f : T \to T$ (regressive in the tree order) so that the inverse image of every point is the union of $<\kappa$ many antichains. More concretely, for every $t$, there is a cardinal $\mu_t<\kappa$ and a function $g_t : f^{-1}\{t\} \to \mu_t$ that is injective on chains. It is easy to see that the existence of these functions implies there is no cofinal branch.

Jensen proved that a special tree on a successor cardinal $\mu^+$ exists iff the weak square principle holds at $\mu$. Todorcevic proved that a strongly inaccessible cardinal $\kappa$ is Mahlo iff there is no special $\kappa$-tree. So there are easy forcings to obtain special trees in these cases that go through these combinatorial theorems. My question concerns obtaining special trees at weakly inaccessible cardinals.

Question: Given a regular cardinal $\kappa$, is there a general ${<}\kappa$-distributive way to force that there is a special $\kappa$-tree, perhaps by bounded approximations?

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Krueger extended Jensen's theorem to arbitrary regular uncountables [1]. In Section 3 of [2], we give a ${<}\kappa$-distributive forcing to add a particular $C$-sequence (witnessing an instance of the proxy principle) whose corresponding $T(\rho_2)$ is a special $\kappa$-Aronszajn tree (by a result from Section 7 of [3]). Assuming $\kappa^{<\kappa}=\kappa$ held in the ground model (update: upon further inspection, this arithmetic hypothesis is surplus. I've just updated the paper to go without it), the said generic extension moreover satisfies that there is a $\kappa$-Souslin tree whose square is special (Section 4 of [2]).

While the $C$-sequence perspective to tree constructions is less transparent (and possibly less inviting), there is a lot of evidence by now suggesting that this is the most fruitful perspective.

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  • $\begingroup$ Thanks! We can force Krueger’s weak square, even a stronger version, with a $<\kappa$-strategically closed forcing, right? Does the version have a name where the $C$-sequence is coherent and the order type function is regressive? $\endgroup$ Commented Nov 16 at 15:33
  • $\begingroup$ Have a look at [2]: we add a coherent sequence whose lower-regressive levels cover a club via a ${<}\kappa$-strategically-closed forcing. This is more than enough to get a special $\kappa$-Aronszajn tree, hence Krueger’s weak square. The definition of lower-regressive levels and related concepts are given by Definition 4.4 of [3]. $\endgroup$ Commented Nov 16 at 17:26
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    $\begingroup$ Thanks. I was just asking if I'm right that the simpler task of just forcing $\square(\kappa,<\kappa)$ by initial segments, to be defined on a generic club $D$, where $\mathrm{ot}(C_\alpha)<\alpha$ for all $\alpha \in D$, is ${<}\kappa$-strategically closed. It seems fine. For a game of length $\gamma$, Player II first pushes the game above $\gamma$ and then plays the sets of lengths of previous moves at limit stages. $\endgroup$ Commented Nov 16 at 20:30

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