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b-scale square principles indecomposable ultrafilter Vanishing levels Reduced Power Diamond for trees super-Souslin tree Singular cardinals combinatorics Subtle tree property Knaster and friends Ostaszewski square Ulam matrix Poset Analytic sets specializable Souslin tree incompactness Fodor-type reflection Microscopic Approach Shelah's Strong Hypothesis AIM forcing Ineffable cardinal full tree coloring number Postprocessing function O-space Precaliber Diamond Fat stationary set Axiom R higher Baire space free Boolean algebra approachability ideal Chromatic number tensor product graph S-Space stick Hedetniemi's conjecture SNR Subadditive Uniformly homogeneous square Rainbow sets Iterated forcing Reflecting stationary set Ascent Path Rock n' Roll middle diamond Kurepa Hypothesis Weakly compact cardinal Prevalent singular cardinals Local Club Condensation. Whitehead Problem Knaster Filter reflection countably metacompact Rado's conjecture HOD weak square Forcing regressive Souslin tree Generalized Clubs Cohen real Successor of Singular Cardinal weak diamond Souslin Tree Small forcing Commutative projection system Jonsson cardinal Distributive tree Strongly Luzin set polarized partition relation Coherent tree Lipschitz reduction Erdos-Hajnal graphs unbounded function Antichain Countryman line GMA C-sequence Amenable C-sequence Absoluteness Constructible Universe Hereditarily Lindelöf space ZFC construction Well-behaved magma Almost-disjoint family reflection principles Greatly Mahlo Fast club Slim tree Chang's conjecture Generalized descriptive set theory Non-saturation Successor of Regular Cardinal Dushnik-Miller strongly bounded groups ccc nonmeager set Uniformization 54G20 Selective Ultrafilter Nonspecial tree Cardinal function transformations stationary reflection Respecting tree Sierpinski's onto mapping principle Strong coloring Singular cofinality Martin's Axiom Universal Sequences positive partition relation Partition Relations Cardinal Invariants Large Cardinals Almost countably chromatic Mandelbrot set Hindman's Theorem Sigma-Prikry Dowker space Luzin set Uniformly coherent Sakurai's Bell inequality Parameterized proxy principle projective Boolean algebra Prikry-type forcing xbox Closed coloring Strongly compact cardinal sap very good scale Square-Brackets Partition Relations PFA(S)[S] free Souslin tree L-space diamond star Commutative cancellative semigroups Singular Density Ramsey theory over partitions Club Guessing Foundations Minimal Walks OCA Almost Souslin weak Kurepa tree Intersection model stationary hitting PFA Was Ulam right club_AD Aronszajn tree Diamond-sharp Subnormal ideal Erdos Cardinal P-Ideal Dichotomy Subtle cardinal Forcing Axioms Open Access
Tag Archives: Square-Brackets Partition Relations
Dushnik-Miller for regular cardinals (part 2)
In this post, we shall provide a proof of Todorcevic’s theorem, that $\mathfrak b=\omega_1$ implies $\omega_1\not\rightarrow(\omega_1,\omega+2)^2$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading
Posted in Blog, Expository
Tagged b-scale, Dushnik-Miller, Partition Relations, Square-Brackets Partition Relations
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CMS Winter Meeting, December 2011
I gave an invited special session talk at the 2011 meeting of the Canadian Mathematical Society. Talk Title: The extent of the failure of Ramsey’s theorem at successor cardinals. Abstract: We shall discuss the results of the following papers: Transforming … Continue reading
Transforming rectangles into squares, with applications to strong colorings
Abstract: It is proved that every singular cardinal $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$. As a … Continue reading
