We introduce the weak partial square principles pλ,κ and p λ,<κ, which combine the ideas of a weak square sequence and a partial square sequence. We construct models in which weak partial square principles fail. The main result of the paper is that λ,κ does not imply λ,<κ. Let λ be a regular uncountable cardinal and let κ ≤ λ be a nonzero cardinal. Recall the square principle λ,κ (respectively ...

This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme, and its strengthening REF(L) (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here L is a language with a distinguished linear order <, and REF(L) consists of the universal closure of formulas of the form ∃x∀y1 < x · · · ∀yn < x φ(y1, · · ·...

Strengthening a result of Amir Leshem [7], we prove that the consistency strength of holding GCH together with definable tree property for all successors of regular cardinals is precisely equal to the consistency strength of existence of proper class many Π 1 reflecting cardinals. Moreover it is proved that if κ is a supercompact cardinal and λ > κ is measurable, then there is a generic extensi...

We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of Pκ(X), where κ ⊆ X. Suppose μ < κ, μ<μ = μ, and κ is supercompact. Then there is a generic extension in which κ = μ++, and for all regular λ ≥ μ++, there are stationarily many N in [H(λ)]μ which are internally club but not internally approachable. Suppose μ is an infinite cardinal. A set N is internally approach...

The games G 1 (κ) and G η <λ(κ) are played by two players in η +complete and max(η+ , λ)-complete Boolean algebras, respectively. For cardinals η, κ such that κ<η = η or κ<η = κ, the (η, κ)-distributive law holds in a Boolean algebra B iff Player 1 does not have a winning strategy in G 1 (κ). Furthermore, for all cardinals κ, the (η,∞)-distributive law holds in B iff Player 1 does not have a wi...

We study the general problem of behaviour continuum function in presence non-supercompact strongly compact cardinals. begin by showing that it is possible to force violations GCH at an arbitrary cardinal using only strong compactness as our initial assumption. This result due third author. then investigate realising Easton functions and above least measurable limit supercompact cardinals starti...

0. Introduction. A stationary subset S of a regular uncountable cardinal κ reflects fully at regular cardinals if for every stationary set T ⊆ κ of higher order consisting of regular cardinals there exists an α ∈ T such that S ∩ α is a stationary subset of α. We prove that the Axiom of Full Reflection which states that every stationary set reflects fully at regular cardinals, together with the ...

We study a two-parameter generalization of the Freese-Nation Property of boolean algebras and its order-theoretic and topological consequences. For every regular infinite κ, the (κ, κ)-FN, the (κ+, κ)-FN, and the κ-FN are known to be equivalent; we show that the family of properties (λ, μ)-FN for λ > μ form a true two-dimensional hierarchy that is robust with respect to coproducts, retracts, an...

1. Introduction. In the problem of bounding the size of codes in compact homogeneous spaces, Del-sarte's polynomial method gives rise to the most powerful universal bounds on codes. Many overviews of the method exist in the literature; see for instance Levenshtein (1998). The purpose of this talk is to present a functional perspective of this method and give some examples. Let X be a compact me...