## Final Report Summary - LAPSCA (Large Properties at Small Cardinals)

Fontanella's work at the Hebrew University of Jerusalem under project LAPSCA was very productive as it resulted in 3 publications for reputable international journals of mathematics.

[1] L. Fontanella and M. Magidor. Reflection of stationary sets and the tree property at the successor of a singular cardinal. To appear in JOURNAL OF SYMBOLIC LOGIC. DOI:10.1017/jsl.2016.13

[2] L. Fontanella and Y. Hayut. Square and Delta reflection. To appear in ANNALS OF PURE AND APPLIED LOGIC. Vol. 167 (8), pp. 663 - 683 (Aug. 2016).

[3] L. Fontanella and P. Matet. Fragments of strong compactness, families of partitions and ideal extensions. FUNDAMENTA MATHEMATICAE, vol. 234, pp. 171- 190 (2016).

The general aim of these papers is to explore several combinatorial principles associated with large cardinals axioms and their interactions; that was precisely the main objective of project LAPSCA.

Paper [1] focuses on two principles, the so-called Reflection of stationary sets and the Tree property. The tree property is the generalisation of König's lemma to uncountable cardinals, while the Reflection of stationary sets for a cardinal kappa is the statement that every stationary subset of kappa is stationary on some uncountable ordinal below kappa. Both are properties of weakly compact cardinals, but can consistently hold even at small cardinals. The literature on both the Tree property and the Reflection of stationary sets is huge, we addressed the question of whether both principles can hold simultaneously at a small cardinal. Several researchers in the past had tried to prove that the Reflection of stationary sets and the Tree property could hold simultaneously at aleph_{omega+1}. Those attempts were unsuccessful and this problem remains open for aleph_{omega+1}, but Fontanella and Magidor were able to prove that these principles can hold simultaneously at another small cardinal of the same kind (a successor of a singular cardinal), namely aleph_{omegaˆ2+1}. More precisely, we proved that assuming the consistency of infinitely many supercompact cardinals, it is possible to build a model of ZFC where aleph_{omegaˆ2+1} satisfies both the Reflection of stationary sets and the Tree property. Fontanella and Magidor's theorem is actually stronger as in our model aleph_{omegaˆ2+1} has an even stronger version of the Reflection of stationary sets called `Delta reflection'. The Delta Reflection was introduced by Magidor and Shelah to study several compactness properties for structures of various kind, including abelian groups, graphs, topological spaces and others. Considered the strength of this principles it was natural to wonder if the Delta Reflection implies directly the Tree property; in the second part of our paper [1] we prove that it is not the case, in fact assuming the consistency of infinitely many supercompact cardinals it is possible to build a model where aleph_{omegaˆ2+1} has the Delta Reflection but the Tree property fails at this cardinal. In conclusion, the two principles are independent but can hold simultaneously at aleph_{omegaˆ2+1}.

Paper [2] concerns the Delta-Reflection, which is a strong form of reflection, and a principle known as Todorcevic's square which is on the contrary an anti-reflection principle. Todorcevic's square for a cardinal is another important combinatorial principle in set theory. This is an anti-reflection principle as it implies the failure of several forms of reflection, for instance Todorcevic's square at a cardinal kappa implies the failure of the simultaneous reflection of stationary sets at kappa (namely, the statement that says that given two stationary subsets S, S' of kappa there exists an ordinal alpha below kappa such that S, S' are both stationary below alpha); it implies also the failure of the tree property at kappa. In this joint paper, Fontanella and Hayut proved that despite the fact that one is a strong reflection principle and the other is a quite strong anti-reflection principle, the Delta-reflection and Todorcevic square can both hold at the same cardinal, namely aleph_{omegaˆ2+1}. The exact statement of this theorem is the following: assuming the consistency of infinitely many supercompact cardinals, it is possible to force a model where both the Delta Reflection and Todorcevic's square hold at aleph_{omegaˆ2+1}. This result has some nice consequences:

- since Todorcevic's square implies the failure of the simultaneous reflection of stationary sets, this result tells us that the Delta-reflection does not imply the simultaneous reflection of stationary sets

- since Todorcevic's square implies the failure of the Tree property, this result tells us gives us another proof that the Delta-reflection does not imply the Tree property (that was the second main result of paper [1])

- since the Delta-reflection implies the failure of the approachability property, this result tells us that Todorcevic's square does not imply the approachability property (unlike Jensen's square)

Moreover, the Delta-reflection was already known to be incompatible with other versions of the square principle such as Jensen's square and the Weak square, so our result gives us the best balance between a reflection principle, the Delta Reflection, and a square-like principle.

Paper [3] explore several combinatorial characterizations of mildly ineffable cardinals. Mildly ineffable cardinals are large cardinals that are associated to the notion of strongly compact cardinal: in fact a cardinal is strongly compact if and only if it is lambda-mildly ineffable for every lambda. In this paper, Fontanella and Matet developed several combinatorial characterizations of mildly ineffable cardinals in terms of partition properties (i.e. generalizations of Ramsey's theorem, the infinite version) and we investigate which of these properties (or weaker versions) can consistently hold at small cardinals and which properties can only be held by large cardinals.

[1] L. Fontanella and M. Magidor. Reflection of stationary sets and the tree property at the successor of a singular cardinal. To appear in JOURNAL OF SYMBOLIC LOGIC. DOI:10.1017/jsl.2016.13

[2] L. Fontanella and Y. Hayut. Square and Delta reflection. To appear in ANNALS OF PURE AND APPLIED LOGIC. Vol. 167 (8), pp. 663 - 683 (Aug. 2016).

[3] L. Fontanella and P. Matet. Fragments of strong compactness, families of partitions and ideal extensions. FUNDAMENTA MATHEMATICAE, vol. 234, pp. 171- 190 (2016).

The general aim of these papers is to explore several combinatorial principles associated with large cardinals axioms and their interactions; that was precisely the main objective of project LAPSCA.

Paper [1] focuses on two principles, the so-called Reflection of stationary sets and the Tree property. The tree property is the generalisation of König's lemma to uncountable cardinals, while the Reflection of stationary sets for a cardinal kappa is the statement that every stationary subset of kappa is stationary on some uncountable ordinal below kappa. Both are properties of weakly compact cardinals, but can consistently hold even at small cardinals. The literature on both the Tree property and the Reflection of stationary sets is huge, we addressed the question of whether both principles can hold simultaneously at a small cardinal. Several researchers in the past had tried to prove that the Reflection of stationary sets and the Tree property could hold simultaneously at aleph_{omega+1}. Those attempts were unsuccessful and this problem remains open for aleph_{omega+1}, but Fontanella and Magidor were able to prove that these principles can hold simultaneously at another small cardinal of the same kind (a successor of a singular cardinal), namely aleph_{omegaˆ2+1}. More precisely, we proved that assuming the consistency of infinitely many supercompact cardinals, it is possible to build a model of ZFC where aleph_{omegaˆ2+1} satisfies both the Reflection of stationary sets and the Tree property. Fontanella and Magidor's theorem is actually stronger as in our model aleph_{omegaˆ2+1} has an even stronger version of the Reflection of stationary sets called `Delta reflection'. The Delta Reflection was introduced by Magidor and Shelah to study several compactness properties for structures of various kind, including abelian groups, graphs, topological spaces and others. Considered the strength of this principles it was natural to wonder if the Delta Reflection implies directly the Tree property; in the second part of our paper [1] we prove that it is not the case, in fact assuming the consistency of infinitely many supercompact cardinals it is possible to build a model where aleph_{omegaˆ2+1} has the Delta Reflection but the Tree property fails at this cardinal. In conclusion, the two principles are independent but can hold simultaneously at aleph_{omegaˆ2+1}.

Paper [2] concerns the Delta-Reflection, which is a strong form of reflection, and a principle known as Todorcevic's square which is on the contrary an anti-reflection principle. Todorcevic's square for a cardinal is another important combinatorial principle in set theory. This is an anti-reflection principle as it implies the failure of several forms of reflection, for instance Todorcevic's square at a cardinal kappa implies the failure of the simultaneous reflection of stationary sets at kappa (namely, the statement that says that given two stationary subsets S, S' of kappa there exists an ordinal alpha below kappa such that S, S' are both stationary below alpha); it implies also the failure of the tree property at kappa. In this joint paper, Fontanella and Hayut proved that despite the fact that one is a strong reflection principle and the other is a quite strong anti-reflection principle, the Delta-reflection and Todorcevic square can both hold at the same cardinal, namely aleph_{omegaˆ2+1}. The exact statement of this theorem is the following: assuming the consistency of infinitely many supercompact cardinals, it is possible to force a model where both the Delta Reflection and Todorcevic's square hold at aleph_{omegaˆ2+1}. This result has some nice consequences:

- since Todorcevic's square implies the failure of the simultaneous reflection of stationary sets, this result tells us that the Delta-reflection does not imply the simultaneous reflection of stationary sets

- since Todorcevic's square implies the failure of the Tree property, this result tells us gives us another proof that the Delta-reflection does not imply the Tree property (that was the second main result of paper [1])

- since the Delta-reflection implies the failure of the approachability property, this result tells us that Todorcevic's square does not imply the approachability property (unlike Jensen's square)

Moreover, the Delta-reflection was already known to be incompatible with other versions of the square principle such as Jensen's square and the Weak square, so our result gives us the best balance between a reflection principle, the Delta Reflection, and a square-like principle.

Paper [3] explore several combinatorial characterizations of mildly ineffable cardinals. Mildly ineffable cardinals are large cardinals that are associated to the notion of strongly compact cardinal: in fact a cardinal is strongly compact if and only if it is lambda-mildly ineffable for every lambda. In this paper, Fontanella and Matet developed several combinatorial characterizations of mildly ineffable cardinals in terms of partition properties (i.e. generalizations of Ramsey's theorem, the infinite version) and we investigate which of these properties (or weaker versions) can consistently hold at small cardinals and which properties can only be held by large cardinals.