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Monotonically far Precaliber Club Guessing coloring number C-sequence Whitehead Problem free Boolean algebra Large Cardinals Knaster and friends Axiom R Shelah's Strong Hypothesis Constructible Universe Prevalent singular cardinals reflection principles weak diamond Partition Relations Luzin set Souslin Tree Knaster Antichain Almost Souslin regressive Souslin tree Subnormal ideal Almost-disjoint family Singular cardinals combinatorics Hindman's Theorem Ramsey theory over partitions OCA Small forcing ZFC construction Erdos-Hajnal graphs Filter reflection Dushnik-Miller Subtle tree property approachability ideal Hedetniemi's conjecture Intersection model square Countryman line Entangled linear order xbox Coherent tree indecomposable filter Ostaszewski square stationary reflection Minimal Walks Diamond for trees Square-Brackets Partition Relations Chang's conjecture Aronszajn tree polarized partition relation Postprocessing function Generalized descriptive set theory Uniformly homogeneous Microscopic Approach Kurepa Hypothesis Sierpinski's onto mapping principle GMA Almost countably chromatic O-space Strongly compact cardinal Non-saturation Successor of Regular Cardinal Erdos Cardinal Generalized Clubs Rado's conjecture Lipschitz reduction weak square transformations sap Subadditive Strongly Luzin set Jonsson cardinal Open Access Universal Sequences SNR free Souslin tree ccc Commutative projection system Fodor-type reflection Local Club Condensation. HOD Fast club Hereditarily Lindelöf space Commutative cancellative semigroups Distributive tree L-space Ineffable cardinal Partition relations for trees Dowker space 54G20 Closed coloring Chromatic number specializable Souslin tree Interval topology on trees square principles PFA countably metacompact strongly bounded groups Vanishing levels Parameterized proxy principle Nonspecial tree Strong coloring diamond star Prikry-type forcing Reflecting stationary set Sigma-Prikry Rock n' Roll Martin's Axiom b-scale Singular Density Selective Ultrafilter Forcing Axioms Ascent Path weak Kurepa tree incompactness Amenable C-sequence Iterated forcing Foundations middle diamond Diamond full tree Greatly Mahlo Cardinal function tensor product graph Rainbow sets AIM forcing Weakly compact cardinal P-Ideal Dichotomy Uniformly coherent PFA(S)[S] Reduced Power Ulam matrix Cardinal Invariants Diamond-sharp Singular cofinality Respecting tree Was Ulam right? Mandelbrot set S-Space Fat stationary set Poset very good scale higher Baire space Analytic sets Uniformization stationary hitting stick nonmeager set Well-behaved magma unbounded function Sakurai's Bell inequality Slim tree Forcing Absoluteness positive partition relation Subtle cardinal perfectly normal Forcing with side conditions Successor of Singular Cardinal super-Souslin tree Ascending path Cohen real club_AD projective Boolean algebra
Tag Archives: 03E45
Square with built-in diamond-plus
Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, Respecting tree, square, xbox
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Putting a diamond inside the square
Abstract. By a 35-year-old theorem of Shelah, $\square_\lambda+\diamondsuit(\lambda^+)$ does not imply square-with-built-in-diamond_lambda for regular uncountable cardinals $\lambda$. Here, it is proved that $\square_\lambda+\diamondsuit(\lambda^+)$ is equivalent to square-with-built-in-diamond_lambda for every singular cardinal $\lambda$. Downloads: Citation information: A. Rinot, Putting a diamond inside … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Diamond, square, Successor of Singular Cardinal
1 Comment
On the consistency strength of the Milner-Sauer conjecture
Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The conjecture is consistent and known … Continue reading
