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