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