Large small sets
A strongly non-Ramsey uncountable graph
It is consistent that there exists a graph X of cardinality such that every graph has an edge coloring with colors in which the induced copies of X (if there are any) are totally multicolored (get all possible colors).
Distinguishing two partition properties of ω1
It is consistent that but .
Vector sets with no repeated differences
We consider the question when a set in a vector space over the rationals, with no differences occurring more than twice, is the union of countably many sets, none containing a difference twice. The answer is “yes” if the set is of size at most , “not” if the set is allowed to be of size . It is consistent that the continuum is large, but the statement still holds for every set smaller than continuum.
Set mappings on generalized linear continua
Some remarks on second category sets
A Ramsey-style extension of a theorem of Erdős and Hajnal
If n, t are natural numbers, μ is an infinite cardinal, G is an n-chromatic graph of cardinality at most μ, then there is a graph X with , |X| = μ⁺, such that every subgraph of X of cardinality < t is n-colorable.
Another proof of a result of Jech and Shelah
Shelah’s pcf theory describes a certain structure which must exist if is strong limit and holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that such partially...
Boolean algebras in which every chain and antichain is countable
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