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