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