Archives
Keywords
Forcing Singular cofinality Generalized descriptive set theory Rock n' Roll transformations Ineffable cardinal Sigma-Prikry Cardinal Invariants Club Guessing Aronszajn tree Uniformly homogeneous Diamond Non-saturation b-scale Chang's conjecture Almost-disjoint family Ostaszewski square Commutative projection system Reduced Power Absoluteness square higher Baire space Local Club Condensation. middle diamond weak square Diamond-sharp OCA free Souslin tree C-sequence super-Souslin tree Dowker space Luzin set Distributive tree Selective Ultrafilter Slim tree Respecting tree Foundations projective Boolean algebra countably metacompact sap Strongly compact cardinal S-Space strongly bounded groups Souslin Tree specializable Souslin tree Prevalent singular cardinals Subnormal ideal Hedetniemi's conjecture Hereditarily Lindelöf space P-Ideal Dichotomy ccc Mandelbrot set Square-Brackets Partition Relations Kurepa Hypothesis stationary reflection Chromatic number PFA positive partition relation Countryman line O-space weak diamond Ascent Path Well-behaved magma Commutative cancellative semigroups regressive Souslin tree Weakly compact cardinal Shelah's Strong Hypothesis PFA(S)[S] Partition Relations polarized partition relation Singular Density Forcing Axioms L-space Constructible Universe Successor of Singular Cardinal Rado's conjecture Strongly Luzin set SNR coloring number stick HOD Knaster approachability ideal tensor product graph GMA Generalized Clubs nonmeager set Dushnik-Miller Parameterized proxy principle Ulam matrix Almost countably chromatic Iterated forcing club_AD Cardinal function Postprocessing function Singular cardinals combinatorics Erdos Cardinal Sakurai's Bell inequality free Boolean algebra Subtle tree property very good scale Lipschitz reduction Precaliber Strong coloring Analytic sets Coherent tree Uniformly coherent Martin's Axiom full tree Fat stationary set Minimal Walks Successor of Regular Cardinal Vanishing levels Sierpinski's onto mapping principle Microscopic Approach Uniformization Greatly Mahlo Fast club indecomposable ultrafilter Filter reflection Knaster and friends weak Kurepa tree Large Cardinals Reflecting stationary set xbox diamond star Antichain Subtle cardinal unbounded function Universal Sequences Almost Souslin Amenable C-sequence Prikry-type forcing stationary hitting Jonsson cardinal Hindman's Theorem Ramsey theory over partitions Whitehead Problem Closed coloring Subadditive Rainbow sets reflection principles Open Access Was Ulam right Small forcing 54G20 Poset Intersection model Diamond for trees square principles Axiom R Cohen real incompactness AIM forcing Nonspecial tree Fodor-type reflection ZFC construction Erdos-Hajnal graphs
Tag Archives: Dushnik-Miller
Dushnik-Miller for regular cardinals (part 3)
Here is what we already know about the Dushnik-Miller theorem in the case of $\omega_1$ (given our earlier posts on the subject): $\omega_1\rightarrow(\omega_1,\omega+1)^2$ holds in ZFC; $\omega_1\rightarrow(\omega_1,\omega+2)^2$ may consistently fail; $\omega_1\rightarrow(\omega_1,\omega_1)^2$ fails in ZFC. In this post, we shall provide … Continue reading
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
27 Comments
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
5 Comments
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
