# $\delta$-strong compactness and generalized strong tree properties

Are there non-trivial equivalent characterizations of $\delta$-strongly compact (and almost strongly compact) cardinals in terms of generalized tree properties?

Recall the definitions as per Joan Bagaria and Menachem Magidor, On $\omega_1$-strongly compact cardinals, JSL 79 (2014), 266-278.

Suppose $\delta > \omega$ is a cardinal. A cardinal $\kappa > \delta$ is $\delta$-strongly compact if for every set $I$, every $\kappa$-complete filter on $I$ can be extended to a $\delta$-complete ultrafilter on $I$. A limit cardinal $\kappa > \omega$ is almost strongly compact if $\kappa$ is $\delta$-strongly compact for every uncountable cardinal $\delta < \kappa$. Note that if $\lambda > \kappa$ and $\kappa$ is $\delta$-strongly compact, then $\lambda$ is also $\delta$-strongly compact, although, apparently serving the interests of Bavel, in the literature there are definitions going in the opposite direction, as noted in Joel's comment.

My question is whether there are localized variants of strong and super tree properties that are equivalent to the parametrized large cardinal concepts. For example, can $\aleph_2$-strong compactness be characterized in terms of a strong tree property hovering around $\aleph_2$? The general project is the search for large cardinal properties that small cardinals may possess, once inaccessibility is dropped.

Notice that by a theorem of Magidor, it is consistent that the first $\omega_1$-strongly compact is also the first measurable, in which case it is strongly compact, so will satisfy the strong tree property. But there is also a model of ZFC in which the first $\omega_1$-strongly compact is singular.

The definitions of those generalized tree properties are given at the end of the question, below.

The context is the results of Di Prisco, Erdös, Jech, Magidor, Tarski, Weiss and Zwicker, who variously prove for the parameter-free concepts:

Theorem

Suppose $\kappa$ is inaccessible.

1. TFAE: (i) $\kappa$ is weakly compact; (ii) $\kappa$ has the tree property; (iii) $\kappa \rightarrow (\kappa)^{2}_{2}$.

2. TFAE: (i) $\kappa$ is strongly compact; (ii) $\kappa$ has the strong tree property.

3. TFAE: (i) $\kappa$ is supercompact; (ii) $\kappa$ has the super tree property.

Now for the definitions of the relevant tree properties, taken from Laura Fontanella, The strong tree property at successors of singular cardinals, JSL 79 (2014), 193-207.

Suppose $\kappa \geq \aleph_2, \lambda \geq \kappa, \lambda \in Ord$. A $(\kappa, \lambda)$-tree is a set $F$ with the properties: (1) for every $f \in F, f:X \rightarrow 2$, for some $X \in [\lambda]^{<\kappa}$; (2) for all $f \in F$, if $X \subseteq dom(f)$, then $f \vert X \in F$; (3) $Lev_X(F) = \lbrace f \in F : dom(f) = X \rbrace \neq \emptyset$ for all $X \in [\lambda]^{<\kappa}$; (4) $\vert Lev_X(F) \vert < \kappa$ for all $X \in [\lambda]^{<\kappa}$.

Note $(\kappa, \lambda)$-trees are not actually trees, despite the terminology: for $f \in Lev_X(F), \lbrace f \vert Y : Y \subseteq X \rbrace$ is not well-ordered in general.

A cofinal branch is a function $b:\lambda \rightarrow 2$ such that $b \vert X \in Lev_X(F)$ for all $X \in [\lambda]^{<\kappa}$.

An $F$-level sequence is a function $D: [\lambda]^{<\kappa} \rightarrow F$ such that $D(X) \in Lev_X(F)$ for every $X \in [\lambda]^{<\kappa}$.

An ineffable branch for an $F$-level sequence $D$ is a cofinal branch $b$ such that $\lbrace X \in [\lambda]^{<\kappa} : b \vert X = D(X) \rbrace$ is stationary.

The cardinal $\kappa$ has the strong tree property if every $(\kappa, \lambda)$-tree has a cofinal branch for every ordinal $\lambda \geq \kappa$.

A cardinal $\kappa$ has the super tree property if for every ordinal $\lambda \geq \kappa$, every $F$-level sequence $D$ in every $(\kappa, \lambda)$-tree has an ineffable branch.

The case of $\aleph_2$ is of interest, much as the classical $\aleph_2$-tree property has been a catalyst of much research and the invention of new forcing (Mitchell); similarly, tree properties at successors of singular cardinals present challenges taken up by many mathematical logicians (Neeman, Unger, Fontanella, Weiss, Cummings, Foreman, Sinapova, Magidor, Shelah, and others).

It is conjectured (by Fontanella) that the successor of a singular limit of supercompact cardinals has the super tree property. In the paper quoted above, she proved that if $\kappa$ is a singular limit of strongly compact cardinals, then $\kappa^+$ has the strong tree property.

• So... what's the question? – Asaf Karagila Jan 16 '15 at 15:13
• @AsafKaragila It seems that he wants to know if those results localize from full strong compactness and full supercompactness to $\delta$-strong compactness and $\delta$-supercompactness. – Joel David Hamkins Jan 16 '15 at 15:46
• Avshalom, it may help if you could state explicitly the strong and super tree properties. – Joel David Hamkins Jan 16 '15 at 15:47
• Avshalom, your definition of $\delta$-strongly compact cardinals has the property that if $\kappa$ is $\delta$-strongly compact, then every larger cardinal $\kappa'>\kappa$ is also $\delta$-strongly compact. Is this what you intend? (I am used to considering $\theta$-strongly compact cardinals only when $\kappa\leq\theta$, but you have it the other way around.) – Joel David Hamkins Jan 16 '15 at 16:00
• @JoelDavidHamkins I've added relevant sources and definitions. – Avshalom Jan 16 '15 at 17:07