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