Category Archives: Blog

What’s next?

I took an offer for a tenure-track position at the Mathematics department of Bar-Ilan University.

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Review: Stevo Todorcevic’s CRM-Fields-PIMS Prize Lecture

After winning the 2012 CRM-Fields-PIMS Prize, Stevo Todorcevic gave a series of talks on his research: at CRM, at PIMS and at the Fields Institute. The director of the Fields Institute asked me to write a short review on Stevo’s … Continue reading

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Prikry Forcing

Recall that the chromatic number of a (symmetric) graph (G,E), denoted Chr(G,E), is the least (possible finite) cardinal κ, for which there exists a coloring c:Gκ such that gEh entails c(g)c(h). Given a forcing notion P, it is … Continue reading

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Shelah’s approachability ideal (part 2)

In a previous post, we defined Shelah’s approachability ideal I[λ]. We remind the reader that a subset Sλ is in I[λ] iff there exists a collection {Dαα<λ}[P(λ)]<λ such that for club many δS, the union … Continue reading

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The uniformization property for 2

Given a subset of a regular uncountable cardinal Sκ, UPS (read: “the uniformization property holds for S”) asserts that for every sequence f=fααS satisfying for all αS: fα is a 2-valued function; dom(fα) is a … Continue reading

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The uniformization property for 2

Given a subset of a regular uncountable cardinal Sκ, UPS (read: “the uniformization property holds for S”) asserts that for every sequence f=fααS satisfying for all αS: fα is a 2-valued function; dom(fα) is a … Continue reading

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The Engelking-Karlowicz theorem, and a useful corollary

Theorem (Engelking-Karlowicz, 1965). For cardinals κλμ2λ, the following are equivalent: λ<κ=λ; there exists a collection of functions, fi:μλi<λ, such that for every X[μ]<κ and every function f:Xλ, there exists some i<λ with ffi. Proof. (2)(1) Suppose … Continue reading

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Kurepa trees and ineffable cardinals

Recall that T is said to be a κ-Kurepa tree if T is a tree of height κ, whose levels Tα has size |α| for co-boundedly many α<κ, and such that the set of branches of T has size >κ. … Continue reading

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Variations on diamond

Jensen’s diamond principle has many equivalent forms. The translation between these forms is often straight-forward, but there is one form whose equivalence to the usual form is somewhat surprising, and Devlin’s translation from one to the other, seems a little … Continue reading

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The P-Ideal Dichotomy and the Souslin Hypothesis

John Krueger is visiting Toronto these days, and in a conversation today, we asked ourselves how do one prove the Abraham-Todorcevic theorem that PID implies SH. Namely, that the next statement implies that there are no Souslin trees: Definition. The … Continue reading

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