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