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Maximal almost disjoint families of functions

Dilip Raghavan (2009)

Fundamenta Mathematicae

We study maximal almost disjoint (MAD) families of functions in ω ω that satisfy certain strong combinatorial properties. In particular, we study the notions of strongly and very MAD families of functions. We introduce and study a hierarchy of combinatorial properties lying between strong MADness and very MADness. Proving a conjecture of Brendle, we show that if c o v ( ℳ ) < , then there no very MAD families. We answer a question of Kastermans by constructing a strongly MAD family from = . Next, we study the...

Minimal predictors in hat problems

Christopher S. Hardin, Alan D. Taylor (2010)

Fundamenta Mathematicae

We consider a combinatorial problem related to guessing the values of a function at various points based on its values at certain other points, often presented by way of a hat-problem metaphor: there are a number of players who will have colored hats placed on their heads, and they wish to guess the colors of their own hats. A visibility relation specifies who can see which hats. This paper focuses on the existence of minimal predictors: strategies guaranteeing at least one player guesses correctly,...

More on tie-points and homeomorphism in ℕ*

Alan Dow, Saharon Shelah (2009)

Fundamenta Mathematicae

A point x is a (bow) tie-point of a space X if X∖x can be partitioned into (relatively) clopen sets each with x in its closure. We denote this as X = A ⋈ x B where A, B are the closed sets which have a unique common accumulation point x. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of βℕ = ℕ* (by Veličković and Shelah Steprans) and in the recent study (by Levy and Dow Techanie) of precisely 2-to-1 maps on ℕ*. In these cases the tie-points have been the unique fixed point...

Multiple gaps

Antonio Avilés, Stevo Todorcevic (2011)

Fundamenta Mathematicae

We study a higher-dimensional version of the standard notion of a gap formed by a finite sequence of ideals of the quotient algebra 𝓟(ω)/fin. We examine different types of such objects found in 𝓟(ω)/fin both from the combinatorial and the descriptive set-theoretic side.

OCA and towers in 𝒫 ( ℕ ) / f i n

Ilijas Farah (1996)

Commentationes Mathematicae Universitatis Carolinae

We shall show that Open Coloring Axiom has different influence on the algebra 𝒫 ( ℕ ) / f i n than on ℕ ℕ / f i n . The tool used to accomplish this is forcing with a Suslin tree.

On a question of Sierpiński

Theodore Slaman (1999)

Fundamenta Mathematicae

There is a set U of reals such that for every analytic set A there is a continuous function f which maps U bijectively to A.

On automorphisms of Boolean algebras embedded in P (ω)/fin

Magdalena Grzech (1996)

Fundamenta Mathematicae

We prove that, under CH, for each Boolean algebra A of cardinality at most the continuum there is an embedding of A into P(ω)/fin such that each automorphism of A can be extended to an automorphism of P(ω)/fin. We also describe a model of ZFC + MA(σ-linked) in which the continuum is arbitrarily large and the above assertion holds true.

On character of points in the Higson corona of a metric space

Taras O. Banakh, Ostap Chervak, Lubomyr Zdomskyy (2013)

Commentationes Mathematicae Universitatis Carolinae

We prove that for an unbounded metric space X , the minimal character 𝗆 χ ( X ˇ ) of a point of the Higson corona X ˇ of X is equal to 𝔲 if X has asymptotically isolated balls and to max { 𝔲 , 𝔡 } otherwise. This implies that under 𝔲 < 𝔡 a metric space X of bounded geometry is coarsely equivalent to the Cantor macro-cube 2 < ℕ if and only if dim ( X ˇ ) = 0 and 𝗆 χ ( X ˇ ) = 𝔡 . This contrasts with a result of Protasov saying that under CH the coronas of any two asymptotically zero-dimensional unbounded metric separable spaces are homeomorphic.

Currently displaying 81 – 100 of 196