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