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