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