Category Archives: Blog

Polychromatic colorings

These are lectures notes of two talks Dani Livne gave in our Infinite Combinatorics seminar. I did not take notes in real-time, hence, all possible mistakes here are due to myself. Recall that a function f:AB is said to … Continue reading

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Universal binary sequences

Notation. Write Q(A):={aAa is finite,a}. Suppose for the moment that we are given a fixed sequence fα:ω2αa, indexed by some set a of ordinals. Then, for every function h:aω and i<ω, we … Continue reading

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Syndetic colorings with applications to S and L

Notation. Write Q(A):={aAa is finite,a}. Definition. An L-space is a regular hereditarily Lindelöf topological space which is not hereditarily separable. Definition. We say that a coloring c:[ω1]2ω is L-syndetic if the following holds. For every uncountable … Continue reading

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Open coloring and the cardinal invariant b

Nik Weaver asked for a direct proof of the fact that Todorcevic’s axiom implies the failure of CH fails. Here goes. Notation. For a set X, we write [X]2 for the set of unordered pairs $\{ \{x,x’\}\mid x,x’\in X, x\neq … Continue reading

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Gabriel Belachsan (14/5/1976 – 20/8/2013)

רק כשעיני סגורות, עולם נגלה לפני

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PFA and the tree property at 2

Recall that a poset T, is said to be a λ+-Aronszajn tree, if it isomorphic to a poset (T,) of the form: T<λ+λ; Write Tα:={σTdom(σ)=α}; for all α<λ+, Tα has size λ, … Continue reading

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A Kurepa tree from diamond-plus

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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The S-space problem, and the cardinal invariant b

Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic … Continue reading

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The S-space problem, and the cardinal invariant b

Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic … Continue reading

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An S-space from a Cohen real

Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In this post, we shall establish the consistency of the existence of such a space. Theorem (Roitman, 1979). Let C=(<ωω,) be the notion of … Continue reading

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