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ccc Universal Sequences polarized partition relation Local Club Condensation. Hindman's Theorem SNR Respecting tree Singular cofinality Postprocessing function Parameterized proxy principle Singular cardinals combinatorics Rado's conjecture b-scale Mandelbrot set coloring number square full tree C-sequence diamond star Fast club Hedetniemi's conjecture super-Souslin tree Minimal Walks Poset HOD Countryman line Knaster and friends Entangled linear order Small forcing countably metacompact Distributive tree specializable Souslin tree Martin's Axiom Kurepa Hypothesis Sakurai's Bell inequality Was Ulam right? OCA Cohen real Slim tree Strongly Luzin set stationary hitting Closed coloring Souslin Tree Aronszajn tree Precaliber Well-behaved magma indecomposable filter Vanishing levels Intersection model Ostaszewski square Diamond-sharp Subtle cardinal strongly bounded groups Knaster Whitehead Problem Dushnik-Miller Uniformization GMA Chromatic number Erdos Cardinal Subnormal ideal weak diamond Square-Brackets Partition Relations Nonspecial tree AIM forcing PFA S-Space Monotonically far PFA(S)[S] Open Access tensor product graph unbounded function Prikry-type forcing Shelah's Strong Hypothesis Selective Ultrafilter Partition Relations L-space Erdos-Hajnal graphs Strongly compact cardinal Almost-disjoint family higher Baire space Generalized descriptive set theory Constructible Universe Strong coloring Uniformly coherent Coherent tree xbox reflection principles Large Cardinals approachability ideal Sierpinski's onto mapping principle sap regressive Souslin tree Fat stationary set Partition relations for trees positive partition relation Commutative cancellative semigroups Jonsson cardinal Forcing Axioms Successor of Regular Cardinal Antichain Forcing O-space Club Guessing Generalized Clubs Ineffable cardinal Forcing with side conditions Analytic sets Uniformly homogeneous Cardinal function P-Ideal Dichotomy Reduced Power Successor of Singular Cardinal stationary reflection 54G20 very good scale square principles Dowker space Cardinal Invariants weak Kurepa tree Fodor-type reflection Amenable C-sequence perfectly normal Ascending path club_AD Diamond Ascent Path Singular Density Ramsey theory over partitions Axiom R Subtle tree property Microscopic Approach Weakly compact cardinal stick Subadditive Reflecting stationary set Chang's conjecture projective Boolean algebra Non-saturation Rainbow sets Filter reflection nonmeager set Commutative projection system Sigma-Prikry Iterated forcing Almost Souslin transformations Almost countably chromatic Foundations incompactness free Boolean algebra Greatly Mahlo free Souslin tree weak square Interval topology on trees ZFC construction Rock n' Roll Luzin set middle diamond Diamond for trees Ulam matrix Lipschitz reduction Prevalent singular cardinals Absoluteness Hereditarily Lindelöf space
Tag Archives: Singular cardinals combinatorics
Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems
Joint work with Menachem Kojman and Juris Steprāns. Abstract. In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in … Continue reading
Sigma-Prikry forcing II: Iteration Scheme
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. In Part I of this series, we introduced a class of notions of forcing which we call $\Sigma$-Prikry, and showed that many of the known Prikry-type notions of forcing that centers … Continue reading
Sigma-Prikry forcing I: The Axioms
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality … Continue reading
More notions of forcing add a Souslin tree
Joint work with Ari Meir Brodsky. Abstract. An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing … Continue reading
Ordinal definable subsets of singular cardinals
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such … Continue reading
Dushnik-Miller for singular cardinals (part 2)
In the first post on this subject, we provided a proof of $\lambda\rightarrow(\lambda,\omega+1)^2$ for every regular uncountable cardinal $\lambda$. In the second post, we provided a proof of $\lambda\rightarrow(\lambda,\omega)^2$ for every singular cardinal $\lambda$, and showed that $\lambda\rightarrow(\lambda,\omega+1)^2$ fails for every … Continue reading
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Tagged Dushnik-Miller, Partition Relations, Singular cardinals combinatorics
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Dushnik-Miller for singular cardinals (part 1)
Continuing the previous post, let us now prove the following. Theorem (Erdos-Dushnik-Miller, 1941). For every singular cardinal λ, we have: $$\lambda\rightarrow(\lambda,\omega)^2.$$ Proof. Suppose that $\lambda$ is a singular cardinal, and $c:[\lambda]^2\rightarrow\{0,1\}$ is a given coloring. For any ordinal $\alpha<\lambda$, denote … Continue reading
On topological spaces of singular density and minimal weight
Abstract: We introduce a weakening of the Generalized Continuum Hypothesis, which we will refer to as the Prevalent Singular cardinals Hypothesis (PSH), and show it implies that every topological space of density and weight $\aleph_{\omega_1}$ is not hereditarily Lindelöf. The assumption … Continue reading
Young Researchers in Set Theory, March 2011
These are the slides of a talk I gave at the Young Researchers in Set Theory 2011 meeting (Königswinter, 21–25 March 2011). Talk Title: Around Jensen’s square principle Abstract: Jensen‘s square principle for a cardinal $\lambda$ asserts the existence of a particular ladder … Continue reading
Workshop on Set Theory and its Applications, February 2007
These are the slides of a talk given at the Workshop on Set Theory and its Applications workshop (Weizmann Institute, February 19, 2007). Talk Title: Nets of spaces having singular density Abstract: The weight of a topological space X is the … Continue reading
