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Indestructible colourings and rainbow Ramsey theorems

Lajos Soukup (2009)

Fundamenta Mathematicae

We show that if a colouring c establishes ω₂ ↛ [(ω₁:ω)]² then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω₂]² → 2 establishing ω₂ ↛ [(ω₁:ω)]₂ such that some colouring g:[ω₁]² → 2 does not embed into c. It is also consistent that 2 ω ₁ is arbitrarily large, and there...

Is the product of ccc spaces a ccc space?

Nina M. Roy (1989)

Publicacions Matemàtiques

In this expository paper it is shown that Martin's Axiom and the negation of the Continuum Hypothesis imply that the product of ccc spaces is a ccc space. The Continuum Hypothesis is then used to construct the Laver-Gavin example of two ccc spaces whose product is not a ccc space.

Level by Level Inequivalence, Strong Compactness, and GCH

Arthur W. Apter (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

We construct three models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In the first two models, below the supercompact cardinal κ, there is a non-supercompact strongly compact cardinal. In the last model, any suitably defined ground model Easton function is realized.

♣-like principles under CH

Winfried Just (2001)

Fundamenta Mathematicae

Some relatives of the Juhász Club Principle are introduced and studied in the presence of CH. In particular, it is shown that a slight strengthening of this principle implies the existence of a Suslin tree in the presence of CH.

Linear orders and MA + ¬wKH

Zoran Spasojević (1995)

Fundamenta Mathematicae

I prove that the statement that “every linear order of size 2 ω can be embedded in ( ω ω , ≪ ) ” is consistent with MA + ¬ wKH.

Locally compact perfectly normal spaces may all be paracompact

Paul B. Larson, Franklin D. Tall (2010)

Fundamenta Mathematicae

We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space...

Lusin sequences under CH and under Martin's Axiom

Uri Abraham, Saharon Shelah (2001)

Fundamenta Mathematicae

Assuming the continuum hypothesis there is an inseparable sequence of length ω₁ that contains no Lusin subsequence, while if Martin's Axiom and ¬ CH are assumed then every inseparable sequence (of length ω₁) is a union of countably many Lusin subsequences.

MAD families and P -points

Salvador García-Ferreira, Paul J. Szeptycki (2007)

Commentationes Mathematicae Universitatis Carolinae

The Katětov ordering of two maximal almost disjoint (MAD) families 𝒜 and ℬ is defined as follows: We say that 𝒜 ≤ K ℬ if there is a function f : ω → ω such that f - 1 ( A ) ∈ ℐ ( ℬ ) for every A ∈ ℐ ( 𝒜 ) . In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called K -uniform if for every X ∈ ℐ ( 𝒜 ) + , we have that 𝒜 | X ≤ K 𝒜 . We prove that CH implies that for every K -uniform MAD family 𝒜 there is a P -point p of ω * such that the set of all Rudin-Keisler predecessors of p is dense in the...

MAD families with strong combinatorial properties

Jörg Brendle, Greg Piper (2007)

Fundamenta Mathematicae

In his paper in Fund. Math. 178 (2003), Miller presented two conjectures regarding MAD families. The first is that CH implies the existence of a MAD family that is also a σ-set. The second is that under CH, there is a MAD family concentrated on a countable subset. These are proved in the present paper.

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