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