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Dowker space Ulam matrix coloring number Successor of Singular Cardinal Fat stationary set Hindman's Theorem positive partition relation Postprocessing function super-Souslin tree Absoluteness specializable Souslin tree PFA(S)[S] Well-behaved magma Cardinal Invariants Local Club Condensation. Commutative cancellative semigroups transformations Rock n' Roll Reduced Power approachability ideal Prikry-type forcing Souslin Tree Open Access OCA Dushnik-Miller Uniformization Diamond Non-saturation Whitehead Problem Almost Souslin Almost countably chromatic strongly bounded groups ccc Large Cardinals Square-Brackets Partition Relations Greatly Mahlo AIM forcing reflection principles Lipschitz reduction full tree Diamond-sharp stationary hitting Constructible Universe Closed coloring Nonspecial tree Partition Relations HOD Filter reflection Diamond for trees stick Jonsson cardinal middle diamond square principles Forcing Axioms very good scale free Boolean algebra Ostaszewski square Cohen real Subtle tree property L-space Selective Ultrafilter Ineffable cardinal Uniformly coherent Minimal Walks Knaster Uniformly homogeneous Martin's Axiom Coherent tree Weakly compact cardinal Parameterized proxy principle Prevalent singular cardinals Rainbow sets Rado's conjecture Foundations square Chromatic number club_AD Strong coloring Singular Density Iterated forcing Precaliber Vanishing levels Reflecting stationary set unbounded function PFA Mandelbrot set SNR nonmeager set Erdos Cardinal free Souslin tree P-Ideal Dichotomy countably metacompact Generalized Clubs polarized partition relation Chang's conjecture Amenable C-sequence diamond star Hereditarily Lindelöf space Subadditive Subnormal ideal Erdos-Hajnal graphs higher Baire space regressive Souslin tree ZFC construction Knaster and friends b-scale indecomposable ultrafilter sap Sigma-Prikry Fodor-type reflection Slim tree Strongly Luzin set Ramsey theory over partitions Subtle cardinal weak Kurepa tree 54G20 stationary reflection tensor product graph Almost-disjoint family Hedetniemi's conjecture Small forcing weak square Shelah's Strong Hypothesis C-sequence O-space Sierpinski's onto mapping principle Kurepa Hypothesis Distributive tree Microscopic Approach Singular cardinals combinatorics Club Guessing GMA Was Ulam right Fast club weak diamond Singular cofinality xbox Poset Ascent Path Forcing Universal Sequences projective Boolean algebra Aronszajn tree Axiom R incompactness Cardinal function Analytic sets Successor of Regular Cardinal Generalized descriptive set theory Luzin set S-Space Sakurai's Bell inequality Antichain
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
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Tagged Rainbow sets, Souslin Tree, Square-Brackets Partition Relations
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Generalizations of Martin’s Axiom and the well-met condition
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
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
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
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
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
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
Open coloring and the cardinal invariant $\mathfrak b$
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
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