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