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Ulam matrix Entangled linear order Hedetniemi's conjecture Cohen real ccc Rock n' Roll C-sequence Prikry-type forcing polarized partition relation higher Baire space Closed coloring Strong coloring Cardinal Invariants L-space very good scale ZFC construction Greatly Mahlo positive partition relation Mandelbrot set Was Ulam right? Uniformly coherent Martin's Axiom Chromatic number Generalized Clubs Uniformly homogeneous Kurepa Hypothesis Sierpinski's onto mapping principle square principles Parameterized proxy principle Ineffable cardinal Selective Ultrafilter club_AD unbounded function Hindman's Theorem Slim tree free Souslin tree Large Cardinals regressive Souslin tree reflection principles weak diamond Reduced Power Ascending path Strongly Luzin set Rainbow sets incompactness Non-saturation Ascent Path Almost Souslin Partition relations for trees sap Iterated forcing Forcing Axioms Postprocessing function Foundations Open Access Successor of Singular Cardinal Cardinal function perfectly normal 54G20 Fat stationary set PFA(S)[S] approachability ideal Almost-disjoint family Countryman line Minimal Walks Nonspecial tree Forcing with side conditions Erdos-Hajnal graphs P-Ideal Dichotomy middle diamond Aronszajn tree projective Boolean algebra Well-behaved magma Knaster Successor of Regular Cardinal Forcing super-Souslin tree S-Space Prevalent singular cardinals Whitehead Problem indecomposable filter Fodor-type reflection Hereditarily Lindelöf space Ramsey theory over partitions Vanishing levels Monotonically far Generalized descriptive set theory transformations Coherent tree Intersection model nonmeager set Absoluteness Commutative projection system Universal Sequences Filter reflection Strongly compact cardinal Weakly compact cardinal Precaliber free Boolean algebra PFA Fast club specializable Souslin tree b-scale Luzin set Square-Brackets Partition Relations strongly bounded groups Singular cardinals combinatorics full tree Diamond for trees Subnormal ideal O-space stationary hitting tensor product graph Chang's conjecture stationary reflection Shelah's Strong Hypothesis HOD xbox Rado's conjecture Poset Microscopic Approach Analytic sets Souslin Tree coloring number countably metacompact Reflecting stationary set Distributive tree Dowker space Almost countably chromatic Sigma-Prikry Respecting tree Axiom R Local Club Condensation. weak Kurepa tree Interval topology on trees Amenable C-sequence Uniformization Singular Density Subtle cardinal square weak square Commutative cancellative semigroups Sakurai's Bell inequality Dushnik-Miller OCA Ostaszewski square Lipschitz reduction GMA Erdos Cardinal diamond star Subadditive Subtle tree property Jonsson cardinal SNR Constructible Universe Partition Relations Diamond-sharp AIM forcing Singular cofinality stick Small forcing Club Guessing Antichain Knaster and friends Diamond
Author Archives: Assaf Rinot
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
Posted in Blog, Expository
Tagged Dushnik-Miller, Partition Relations, Singular cardinals combinatorics
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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
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
Dushnik-Miller for regular cardinals (part 1)
This is the first out of a series of posts on the following theorem. Theorem (Erdos-Dushnik-Miller, 1941). For every infinite cardinal $\lambda$, we have: $$\lambda\rightarrow(\lambda,\omega)^2.$$ Namely, for any coloring $c:[\lambda]^2\rightarrow\{0,1\}$ there exists either a subset $A\subseteq \lambda$ of order-type $\lambda$ with … Continue reading
The order-type of clubs in a square sequence
Recall Jensen’s notion of square: Definition (Jensen): For an infinite cardinal $\lambda$, $\square_\lambda$ asserts the existence of a sequence $\overrightarrow C=\left\langle C_\alpha\mid\alpha\in\text{acc}(\lambda^+)\right\rangle$ such that for every limit $\alpha<\lambda^+$: $C_\alpha$ is a club subset of $\alpha$ of order-type $\le\lambda$; if $\beta\in\text{acc}(C_\alpha)$, … Continue reading
Music Video: “Wide Open” by Jenny Mayhem
Did you notice the toolbar at the bottom of my posts? e.g.:
Posted in Blog, OffMath
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The search for diamonds
Abstract: This is a review I wrote for the Bulletin of Symbolic Logic on the following papers: Saharon Shelah, Middle Diamond, Archive for Mathematical Logic, vol. 44 (2005), pp. 527–560. Saharon Shelah, Diamonds, Proceedings of the American Mathematical Society, vol. … Continue reading
Posted in Publications, Reviews, Squares and Diamonds
Tagged Diamond, middle diamond, weak diamond, weak square
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Aspects of singular cofinality
Abstract. We study properties of closure operators of singular cofinality, and introduce several ZFC sufficient and equivalent conditions for the existence of antichain sequences in posets of singular cofinality. We also notice that the Proper Forcing Axiom implies the Milner-Sauer … Continue reading
Jensen’s diamond principle and its relatives
This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … Continue reading
