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countably metacompact PFA(S)[S] free Boolean algebra projective Boolean algebra Closed coloring Sigma-Prikry SNR Vanishing levels Non-saturation polarized partition relation Erdos-Hajnal graphs ZFC construction Almost-disjoint family L-space Reflecting stationary set Foundations Nonspecial tree approachability ideal Whitehead Problem stationary hitting strongly bounded groups Uniformization Chang's conjecture Almost Souslin super-Souslin tree Rado's conjecture HOD C-sequence Strong coloring Distributive tree Knaster Greatly Mahlo Prikry-type forcing Axiom R Subtle cardinal Kurepa Hypothesis coloring number Parameterized proxy principle Hedetniemi's conjecture Souslin Tree reflection principles Large Cardinals Generalized Clubs O-space very good scale Singular cofinality free Souslin tree Filter reflection ccc Open Access Ascent Path Fodor-type reflection Antichain unbounded function Prevalent singular cardinals Minimal Walks Ostaszewski square Uniformly coherent Dushnik-Miller Iterated forcing square principles Analytic sets specializable Souslin tree Hindman's Theorem Fast club Coherent tree Club Guessing square Successor of Regular Cardinal weak Kurepa tree Generalized descriptive set theory 54G20 Fat stationary set Commutative cancellative semigroups Square-Brackets Partition Relations Diamond sap Mandelbrot set S-Space Was Ulam right Luzin set Singular cardinals combinatorics transformations PFA Absoluteness Poset Dowker space Rainbow sets regressive Souslin tree Subtle tree property weak square Partition Relations Forcing Precaliber Universal Sequences Chromatic number Forcing Axioms Knaster and friends Successor of Singular Cardinal Ineffable cardinal tensor product graph Reduced Power Singular Density xbox Cohen real stick Sierpinski's onto mapping principle higher Baire space Selective Ultrafilter GMA Jonsson cardinal Local Club Condensation. Sakurai's Bell inequality Postprocessing function Cardinal function b-scale Hereditarily Lindelöf space Subnormal ideal Microscopic Approach nonmeager set Lipschitz reduction Rock n' Roll diamond star Erdos Cardinal weak diamond Subadditive positive partition relation P-Ideal Dichotomy Aronszajn tree club_AD Small forcing Well-behaved magma incompactness middle diamond Slim tree full tree AIM forcing Shelah's Strong Hypothesis Cardinal Invariants Diamond-sharp Strongly Luzin set Amenable C-sequence indecomposable ultrafilter Diamond for trees Constructible Universe Ramsey theory over partitions Weakly compact cardinal Ulam matrix OCA Martin's Axiom Uniformly homogeneous stationary reflection Almost countably chromatic
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Jones’ theorem on the cardinal invariant $\mathfrak p$
This post continues the study of the cardinal invariant $\mathfrak p$. We refer the reader to a previous post for all the needed background. For ordinals $\alpha,\alpha_0,\alpha_1,\beta,\beta_0,\beta_1$, the polarized partition relation $$\left(\begin{array}{c}\alpha\\\beta\end{array}\right)\rightarrow\left(\begin{array}{cc}\alpha_0&\alpha_1\\\beta_0&\beta_1\end{array}\right)$$ asserts that for every coloring $f:\alpha\times\beta\rightarrow 2$, (at least) … Continue reading
Posted in Blog, Expository
Tagged polarized partition relation, Sierpinski's onto mapping principle
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