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