We provide a lower bound for the consistency strength of the hypothesis proposed by S. Kamo: ∃κ > א0 s(κ) ≥ κ. 0. Introduction. In [3] ,[10] cardinal invariants on ω are generalized for uncountable regular cardinals and it is shown that some of their properties still hold true for their uncountable counterparts.However,the role of the splitting number s(κ) (for the definition see below) changes...

We study several cardinal, and ordinal–valued functions that are relatives of Hanf numbers. Let κ be an infinite cardinal, and let T ⊆ Lκ+,ω be a theory of cardinality ≤ κ, and let γ be an ordinal ≥ κ . For example we look at 1. μ∗T (γ, κ) := min{μ ∗ : ∀φ ∈ L∞,ω, with rk(φ) < γ, if T has the (φ, μ∗)-order property then there exists a formula φ ′ (x; y) ∈ Lκ+,ω, such that for every χ ≥ κ, T has ...

We present a technique for destroying stationary subsets of Pκκ using partial square sequences. We combine this method with Gitik’s poset for changing the cofinality of a cardinal without adding bounded sets to prove a variety of consistency results concerning saturated ideals and the set S(κ, κ+). In this paper we continue our study of consistency results concerning the set S(κ, κ) = {a ∈ Pκκ ...

Every set of ordinals of cardinality κ in a Prikry extension with a measure over κ contains an old set of arbitrary large cardinality below κ, and, actually, it can be split into countably many old sets. What about sets bigger cardinalities? Clearly, any set of ordinals in a forcing extension of a regular cardinality above the cardinality of the forcing used, contains an old set of the same car...

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 present an example of two connected graphs for which the connectivity of the Cartesian product of the graphs is strictly greater than the sum of the connectivities of the factor graphs. This clarifies an issue from the literature. We then find necessary and sufficient conditions for the connectivity of the Cartesian product of the graphs to be equal to the sum of the connectivities of the in...

If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ+,∞)-distributive and λ is 2λ supercompact, then by [3, Theorem 5], {δ < κ | δ is δ+ strongly compact yet δ isn’t δ+ supercompact} must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in whic...

Questions regarding the continuity in κ of SLEκ traces and maps appear very naturally study SLE. In order to first question, we consider a natural coupling SLE traces: for different values use same Brownian motion. It is assume that with probability one, depends continuously on κ. rather easy show continuous Carathéodory sense, but showing are uniform sense much harder. this note given sequence...

Configurations of type (κ +1)κ give rise to κ–regular simple graphs via configuration graphs. On the other hand, neighbourhood geometries of C4–free κ–regular simple graphs on κ 2 + 1 vertices turn out to be configurations of type (κ + 1)κ. We investigate which configurations of type (κ +1)κ are equal or isomorphic to the neighbourhood geometry of their configuration graph and conversely. We cl...