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Whitehead Problem SNR Subtle cardinal Universal Sequences Erdos Cardinal Successor of Regular Cardinal Uniformly homogeneous ccc higher Baire space Jonsson cardinal Large Cardinals Open Access Lipschitz reduction Forcing Forcing Axioms Local Club Condensation. Uniformization nonmeager set Countryman line specializable Souslin tree Slim tree Microscopic Approach Ulam matrix reflection principles Shelah's Strong Hypothesis Square-Brackets Partition Relations stationary reflection Knaster Rock n' Roll Souslin Tree perfectly normal full tree diamond star weak Kurepa tree coloring number Well-behaved magma Prikry-type forcing polarized partition relation incompactness Fat stationary set Generalized Clubs transformations Parameterized proxy principle Sierpinski's onto mapping principle Fast club unbounded function Dowker space Amenable C-sequence Hereditarily Lindelöf space Absoluteness very good scale Respecting tree HOD GMA Diamond-sharp Diamond S-Space AIM forcing Nonspecial tree Forcing with side conditions Sakurai's Bell inequality Minimal Walks Hindman's Theorem Poset Was Ulam right? PFA Dushnik-Miller xbox free Boolean algebra Prevalent singular cardinals Intersection model Cohen real weak diamond Ascent Path indecomposable filter Selective Ultrafilter Entangled linear order Monotonically far Kurepa Hypothesis L-space Almost-disjoint family Uniformly coherent Rainbow sets Greatly Mahlo Interval topology on trees ZFC construction Fodor-type reflection b-scale Reflecting stationary set Reduced Power Mandelbrot set Ineffable cardinal Commutative projection system Subtle tree property Hedetniemi's conjecture Martin's Axiom Cardinal Invariants Subnormal ideal Partition Relations Erdos-Hajnal graphs projective Boolean algebra C-sequence Vanishing levels strongly bounded groups Subadditive weak square Generalized descriptive set theory Precaliber Distributive tree stationary hitting tensor product graph O-space Small forcing OCA Luzin set Postprocessing function Sigma-Prikry Analytic sets Strong coloring Ostaszewski square Aronszajn tree Ascending path 54G20 middle diamond countably metacompact Cardinal function Strongly Luzin set Almost Souslin super-Souslin tree Chromatic number square Singular cofinality Singular Density Rado's conjecture Diamond for trees P-Ideal Dichotomy club_AD square principles regressive Souslin tree Foundations Club Guessing Singular cardinals combinatorics Filter reflection Weakly compact cardinal positive partition relation Almost countably chromatic Constructible Universe Commutative cancellative semigroups Chang's conjecture stick Ramsey theory over partitions free Souslin tree Knaster and friends Closed coloring Iterated forcing Successor of Singular Cardinal sap Strongly compact cardinal Partition relations for trees Axiom R PFA(S)[S] approachability ideal Antichain Coherent tree Non-saturation
Category Archives: Expository
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
Shelah’s solution to Whitehead’s problem
Whitehead problem notes in hebrew : Table of contents Chapter 0 Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 References
