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