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Luzin set diamond star Ostaszewski square Weakly compact cardinal Forcing Foundations Subadditive Erdos-Hajnal graphs AIM forcing Subnormal ideal Uniformization unbounded function Ramsey theory over partitions countably metacompact approachability ideal Hereditarily Lindelöf space Partition Relations super-Souslin tree 54G20 Was Ulam right Reduced Power Cardinal function Forcing Axioms Microscopic Approach Kurepa Hypothesis Strong coloring Uniformly homogeneous Amenable C-sequence Filter reflection Shelah's Strong Hypothesis Almost-disjoint family square Singular Density Fast club Ascent Path Rado's conjecture Rock n' Roll Ulam matrix Subtle cardinal ccc ZFC construction Non-saturation Club Guessing Distributive tree Hedetniemi's conjecture incompactness Dushnik-Miller Cohen real Analytic sets indecomposable ultrafilter polarized partition relation free Boolean algebra sap weak square OCA transformations Knaster Commutative cancellative semigroups Strongly Luzin set nonmeager set Prevalent singular cardinals specializable Souslin tree Aronszajn tree Precaliber Slim tree Coherent tree Greatly Mahlo O-space HOD Axiom R Martin's Axiom Large Cardinals coloring number Singular cardinals combinatorics Hindman's Theorem regressive Souslin tree middle diamond b-scale Selective Ultrafilter Poset Lipschitz reduction S-Space stationary hitting Diamond for trees Diamond-sharp PFA L-space Reflecting stationary set Sierpinski's onto mapping principle Well-behaved magma Sakurai's Bell inequality Knaster and friends Local Club Condensation. club_AD Singular cofinality Nonspecial tree Whitehead Problem Successor of Singular Cardinal free Souslin tree Diamond Almost countably chromatic Fodor-type reflection Generalized Clubs Successor of Regular Cardinal Souslin Tree Closed coloring xbox tensor product graph Fat stationary set Antichain Constructible Universe Ineffable cardinal Vanishing levels reflection principles PFA(S)[S] Erdos Cardinal full tree Absoluteness Uniformly coherent Dowker space Postprocessing function Rainbow sets Subtle tree property stick Parameterized proxy principle Universal Sequences Almost Souslin weak diamond Open Access very good scale Iterated forcing Cardinal Invariants square principles Minimal Walks Square-Brackets Partition Relations Chang's conjecture Chromatic number Generalized descriptive set theory positive partition relation Prikry-type forcing C-sequence Jonsson cardinal SNR Mandelbrot set strongly bounded groups P-Ideal Dichotomy Sigma-Prikry stationary reflection higher Baire space GMA projective Boolean algebra Small forcing
Blog Archives
The eightfold way
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … 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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Square principles
Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial. Definition. A square principle … Continue reading
Prikry Forcing
Recall that the chromatic number of a (symmetric) graph $(G,E)$, denoted $\text{Chr}(G,E)$, is the least (possible finite) cardinal $\kappa$, for which there exists a coloring $c:G\rightarrow\kappa$ such that $gEh$ entails $c(g)\neq c(h)$. Given a forcing notion $\mathbb P$, it is … Continue reading