The productivity of the κ-chain condition, where κ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of κ-cc posets whose squares are not κ-cc were constructed by Laver, Galvin, Roitman and Fleissner. Later, ZFC examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which κ = ℵ 2 , was resolved by Shelah in 1997.In this work, we obtain analogous results regarding the infinite productivity of strong chain conditions, such as the Knaster property. Among other results, for any successor cardinal κ, we produce a ZFC example of a poset with precaliber κ whose ω th power is not κ-cc. To do so, we carry out a systematic study of colorings satisfying a strong unboundedness condition. We prove a number of results indicating circumstances under which such colorings exist, in particular focusing on cases in which these colorings are moreover closed.
Abstract. We propose a parameterized proxy principle from which κ-Souslin trees with various additional features can be constructed, regardless of the identity of κ. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent κ-Souslin tree that applies also for κ inaccessible.We then carry out a systematic study of the consistency of instances of the proxy principle, distinguished by the vector of parameters serving as its input. Among other things, it will be shown that all known ♦-based constructions of κ-Souslin trees may be redirected through this new proxy principle.
The history of productivity of the κ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal κ > ℵ 1 , the principle (κ) is equivalent to the existence of a certain strong coloring c : [κ] 2 → κ for which the family of fibers T (c) is a nonspecial κ-Aronszajn tree.The theorem follows from an analysis of a new characteristic function for walks on ordinals, and implies in particular that if the κ-chain condition is productive for a given regular cardinal κ > ℵ 1 , then κ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if κ is a weakly compact cardinal, then the κ-chain condition is productive. §1. Introduction.
The chain condition.Two elements x, y in a poset P, ≤ are said to be compatible, if there exists some z ∈ P such that x ≤ z and y ≤ z. A subset A ⊆ P is an antichain if all of its elements are pairwise incompatible. A poset is said to satisfy the κ-cc (short for κ-chain condition), if all of its antichains are of size < κ. A topological space X = X, is said to satisfy the κ-cc if the poset \ {∅}, ⊇ satisfies the κ-cc, that is, if any collection of pairwise disjoint open sets in X has size < κ. A Boolean algebra B is said to satisfy the κ-cc if any collection of pairwise disjoint elements of B has size < κ.
Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that together with 2λ = λ+ implies ⋄S for every S ⊆ λ+ that reflects stationarily often.In this paper, for a set S ⊆ λ+, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; λ]. We say that the ideal is fat if it contains a stationary set. It is proved:1. if I[S; λ] is fat, then NSλ + ∣ S is non-saturated;2. if I[S; λ] is fat and 2λ = λ+, then ⋄S holds;3. implies that I[S; λ] is fat for every S ⊆ λ+ that reflects stationarily often;4. it is relatively consistent with the existence of a supercompact cardinal that fails, while I[S; λ] is fat for every stationary S ⊆ λ+ that reflects stationarily often.The stronger principle is studied as well.
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