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