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