Category Archives: Blog

An inconsistent form of club guessing

In this post, we shall present an answer (due to P. Larson) to a question by A. Primavesi concerning a certain strong form of club guessing. We commence with recalling Shelah’s concept of club guessing. Concept (Shelah). Given a regular … Continue reading

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c.c.c. forcing without combinatorics

In this post, we shall discuss a short paper by Alan Mekler from 1984, concerning a non-combinatorial verification of the c.c.c. property for forcing notions. Recall that a notion of forcing P is said to satisfy the c.c.c. iff … Continue reading

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Dushnik-Miller for singular cardinals (part 2)

In the first post on this subject, we provided a proof of λ(λ,ω+1)2 for every regular uncountable cardinal λ. In the second post, we provided a proof of λ(λ,ω)2 for every singular cardinal λ, and showed that λ(λ,ω+1)2 fails for every … Continue reading

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Dushnik-Miller for regular cardinals (part 2)

In this post, we shall provide a proof of Todorcevic’s theorem, that b=ω1 implies ω1(ω1,ω+2)2. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading

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Dushnik-Miller for singular cardinals (part 1)

Continuing the previous post, let us now prove the following. Theorem (Erdos-Dushnik-Miller, 1941). For every singular cardinal λ, we have: λ(λ,ω)2. Proof. Suppose that λ is a singular cardinal, and c:[λ]2{0,1} is a given coloring. For any ordinal α<λ, denote … Continue reading

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Dushnik-Miller for regular cardinals (part 1)

This is the first out of a series of posts on the following theorem. Theorem (Erdos-Dushnik-Miller, 1941). For every infinite cardinal λ, we have: λ(λ,ω)2. Namely, for any coloring c:[λ]2{0,1} there exists either a subset Aλ of order-type λ with … Continue reading

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The order-type of clubs in a square sequence

Recall Jensen’s notion of square: Definition (Jensen): For an infinite cardinal λ, ◻λ asserts the existence of a sequence C=Cααacc(λ+) such that for every limit α<λ+: Cα is a club subset of α of order-type λ; if βacc(Cα), … Continue reading

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Music Video: “Wide Open” by Jenny Mayhem

Did you notice the toolbar at the bottom of my posts? e.g.:

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James Earl Baumgartner

Sad news: Jim Baumgartner passed away. See here.

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