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Forcing Axioms Ineffable cardinal club_AD Martin's Axiom ccc Commutative cancellative semigroups positive partition relation Knaster Coherent tree Partition Relations Ulam matrix weak diamond nonmeager set Kurepa Hypothesis Ramsey theory over partitions stick Luzin set reflection principles Successor of Singular Cardinal regressive Souslin tree Cohen real Constructible Universe higher Baire space Absoluteness Entangled linear order Mandelbrot set Uniformization Successor of Regular Cardinal C-sequence Strongly Luzin set Rado's conjecture L-space Cardinal function Subadditive Antichain Rainbow sets Knaster and friends Prikry-type forcing Uniformly coherent Iterated forcing SNR specializable Souslin tree Generalized descriptive set theory stationary reflection Cardinal Invariants Subtle cardinal Ascent Path Jonsson cardinal Amenable C-sequence Sakurai's Bell inequality Almost countably chromatic Weakly compact cardinal P-Ideal Dichotomy Large Cardinals Well-behaved magma Commutative projection system very good scale Nonspecial tree S-Space free Souslin tree Slim tree Monotonically far Forcing Subtle tree property Subnormal ideal Distributive tree Reflecting stationary set Erdos-Hajnal graphs Almost-disjoint family countably metacompact Small forcing AIM forcing Singular cardinals combinatorics Sierpinski's onto mapping principle Rock n' Roll Greatly Mahlo Square-Brackets Partition Relations unbounded function projective Boolean algebra square principles Open Access Chang's conjecture PFA Was Ulam right? Aronszajn tree b-scale Club Guessing Dushnik-Miller Non-saturation xbox approachability ideal Hindman's Theorem Vanishing levels Dowker space Strong coloring ZFC construction Respecting tree middle diamond OCA Diamond for trees stationary hitting free Boolean algebra Intersection model Chromatic number Microscopic Approach weak Kurepa tree Ostaszewski square Diamond Sigma-Prikry coloring number Axiom R Reduced Power incompactness Generalized Clubs 54G20 sap transformations Universal Sequences Closed coloring strongly bounded groups PFA(S)[S] perfectly normal Forcing with side conditions Parameterized proxy principle Uniformly homogeneous polarized partition relation indecomposable filter Fat stationary set Foundations Partition relations for trees Hedetniemi's conjecture Singular Density Filter reflection O-space GMA Interval topology on trees weak square square HOD Precaliber Analytic sets Fast club Strongly compact cardinal Almost Souslin Countryman line Whitehead Problem Fodor-type reflection Souslin Tree tensor product graph full tree Diamond-sharp Poset Hereditarily Lindelöf space Ascending path Lipschitz reduction diamond star Local Club Condensation. Singular cofinality Shelah's Strong Hypothesis Minimal Walks Erdos Cardinal super-Souslin tree Selective Ultrafilter Postprocessing function Prevalent singular cardinals
Category Archives: Publications
More notions of forcing add a Souslin tree
Joint work with Ari Meir Brodsky. Abstract. An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing … Continue reading
Ordinal definable subsets of singular cardinals
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such … Continue reading
Higher Souslin trees and the GCH, revisited
Abstract. It is proved that for every uncountable cardinal $\lambda$, GCH+$\square(\lambda^+)$ entails the existence of a $\text{cf}(\lambda)$-complete $\lambda^+$-Souslin tree. In particular, if GCH holds and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in Godel’s constructible universe, improving … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, Open Access, regressive Souslin tree, Souslin Tree, square, Weakly compact cardinal, xbox
16 Comments
A microscopic approach to Souslin-tree constructions. Part I
Joint work with Ari Meir Brodsky. Abstract. We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E65, 05C05, Coherent tree, Diamond, Microscopic Approach, Parameterized proxy principle, Slim tree, Souslin Tree, square, xbox
5 Comments
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
1 Comment
Reduced powers of Souslin trees
Joint work with Ari Meir Brodsky. Abstract. We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed … Continue reading
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
Same Graph, Different Universe
Abstract. May the same graph admit two different chromatic numbers in two different universes? how about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Godel’s constructible … Continue reading
Posted in Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, approachability ideal, Chromatic number, Constructible Universe, Forcing, Ostaszewski square
10 Comments
Chain conditions of products, and weakly compact cardinals
Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading
Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
2 Comments
Complicated colorings
Abstract. If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^\lambda_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. (Recall that $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ asserts the existence of a coloring $d:[\lambda]^2\rightarrow\lambda$ such that for any family $\mathcal A\subseteq[\lambda]^{<\kappa}$ of size $\lambda$, consisting of pairwise … Continue reading
Posted in Partition Relations, Publications
Tagged Minimal Walks, Open Access, Square-Brackets Partition Relations
2 Comments
