# 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 $f:A\rightarrow B$ is said to … Continue reading

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## Universal binary sequences

Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Suppose for the moment that we are given a fixed sequence $\langle f_\alpha:\omega\rightarrow2\mid \alpha\in a\rangle$, indexed by some set $a$ of ordinals. Then, for every function $h:a\rightarrow\omega$ and $i<\omega$, we … Continue reading

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## Syndetic colorings with applications to S and L

Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Definition. An L-space is a regular hereditarily Lindelöf topological space which is not hereditarily separable. Definition. We say that a coloring $c:[\omega_1]^2\rightarrow\omega$ is L-syndetic if the following holds. For every uncountable … Continue reading

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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 $\{ \{x,x’\}\mid x,x’\in X, x\neq … Continue reading Posted in Blog, Expository | Tagged , | 13 Comments ## Gabriel Belachsan (14/5/1976 – 20/8/2013) רק כשעיני סגורות, עולם נגלה לפני Posted in Blog, OffMath | Tagged | Leave a comment ## PFA and the tree property at$\aleph_2$Recall that a poset$\langle T,\le\rangle$is said to be a$\lambda^+$-Aronszajn tree, if it isomorphic to a poset$(\mathcal T,\subseteq)$of the form:$\emptyset\in \mathcal T\subseteq{}^{<\lambda^+}\lambda$; Write$\mathcal T_\alpha:=\{\sigma\in\mathcal T\mid \text{dom}(\sigma)=\alpha\}$; for all$\alpha<\lambda^+$,$\mathcal T_\alpha$has size$\le\lambda$, … Continue reading Posted in Blog, Expository | Tagged , | 5 Comments ## A Kurepa tree from diamond-plus Recall that$T$is said to be a$\kappa$-Kurepa tree if$T$is a tree of height$\kappa$, whose levels$T_\alpha$has size$\le|\alpha|$for co-boundedly many$\alpha<\kappa$, and such that the set of branches of$T$has size$>\kappa$. … Continue reading Posted in Blog, Expository | | Leave a comment ## The S-space problem, and the cardinal invariant$\mathfrak b$Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic … Continue reading Posted in Blog, Expository | Tagged , | Leave a comment ## An$S$-space from a Cohen real Recall that an$S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In this post, we shall establish the consistency of the existence of such a space. Theorem (Roitman, 1979). Let$\mathbb C=({}^{<\omega}\omega,\subseteq)$be the notion of … Continue reading Posted in Blog, Expository | Tagged , | 5 Comments ## Forcing with a Souslin tree makes$\mathfrak p=\omega_1\$

I was meaning to include a proof of Farah’s lemma in my previous post, but then I realized that the slick proof assumes some background which may worth spelling out, first. Therefore, I am dedicating a short post for a … Continue reading

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