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**1 - 2**of**2**### Concerning problems about cardinal invariants on Boolean algebras

, 2008

"... The purpose of these notes is to describe the progress made on the 97 open problems formulated in the book Cardinal invariants on Boolean algebras, hereafter denoted by [CI]. Although we assume acquaintance with that book, we give some background for many problems, and state without proof most resul ..."

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The purpose of these notes is to describe the progress made on the 97 open problems formulated in the book Cardinal invariants on Boolean algebras, hereafter denoted by [CI]. Although we assume acquaintance with that book, we give some background for many problems, and state without proof most results relevant to the problems, as far as the author knows. Many of the problems have been solved, at least partially, by Saharon Shelah. For the unpublished papers of Shelah mentioned in the references, see his archive, which can currently be obtained at the URL

### ON A PROBLEM OF STEVE KALIKOW

"... Abstract. The Kalikow problem for a pair (λ, κ) of cardinal numbers, λ> κ (in particular κ = 2) is whether we can map the family of ω– sequences from λ to the family of ω–sequences from κ in a very continuous manner. Namely, we demand that for η, ν ∈ ωλ we have: η, ν are almost equal if and only ..."

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Abstract. The Kalikow problem for a pair (λ, κ) of cardinal numbers, λ> κ (in particular κ = 2) is whether we can map the family of ω– sequences from λ to the family of ω–sequences from κ in a very continuous manner. Namely, we demand that for η, ν ∈ ωλ we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for ℵω but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants. (590) revision:1999-11-12 modified:1999-11-12 In the present paper we are interested in the following property of pairs of cardinal numbers: Definition 0.1. Let λ, κ be cardinals. We say that the pair (λ, κ) has the Kalikow property (and then we write KL(λ, κ)) if