Category Archives: Expository

Prolific Souslin trees

In a paper from 1971, Erdos and Hajnal asked whether (assuming CH) every coloring witnessing $\aleph_1\nrightarrow[\aleph_1]^2_3$ has a rainbow triangle. The negative solution was given in a 1975 paper by Shelah, and the proof and relevant definitions may be found … Continue reading

Recall that Martin’s Axiom asserts that for every partial order $\mathbb P$ satisfying c.c.c., and for any family $\mathcal D$ of $<2^{\aleph_0}$ many dense subsets of $\mathbb P$, there exists a directed subset $G$ of $\mathbb P$ such that $G\cap … Continue reading Posted in Blog, Expository | Tagged , , | Leave a comment Many diamonds from just one Recall Jensen’s diamond principle over a stationary subset$S$of a regular uncountable cardinal$\kappa$: there exists a sequence$\langle A_\alpha\mid \alpha\in S \rangle$such that$\{\alpha\in S\mid A\cap\alpha=A_\alpha\}$is stationary for every$A\subseteq\kappa$. Equivalently, there exists a sequence$\langle … Continue reading

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Square principles

Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial. Definition. A square principle … Continue reading

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Partitioning the club guessing

In a recent paper, I am making use of the following  fact. Theorem (Shelah, 1997). Suppose that $\kappa$ is an accessible cardinal (i.e., there exists a cardinal $\theta<\kappa$ such that $2^\theta\ge\kappa)$. Then there exists a sequence $\langle g_\delta:C_\delta\rightarrow\omega\mid \delta\in E^{\kappa^+}_\kappa\rangle$ … Continue reading

Walk on countable ordinals: the characteristics

In this post, we shall present a few aspects of the method of walk on ordinals (focusing on countable ordinals), record its characteristics, and verify some of their properties. All definitions and results in this post are due to Todorcevic. … Continue reading

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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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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 $\{ \{x,x’\}\mid x,x’\in X, x\neq … Continue reading Posted in Blog, Expository | Tagged , | 13 Comments PFA and the tree property at$\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

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