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reflection principles Countryman line Strongly Luzin set free Boolean algebra stationary hitting stationary reflection polarized partition relation Commutative projection system Greatly Mahlo Was Ulam right? square Prikry-type forcing Almost-disjoint family full tree Sigma-Prikry PFA incompactness perfectly normal Square-Brackets Partition Relations Almost Souslin Ulam matrix Antichain Successor of Regular Cardinal Minimal Walks Ascent Path Luzin set Forcing Dushnik-Miller GMA HOD weak diamond Vanishing levels Weakly compact cardinal Cardinal Invariants Coherent tree stick Strongly compact cardinal O-space Strong coloring Partition relations for trees Reduced Power Erdos-Hajnal graphs projective Boolean algebra Poset Local Club Condensation. Small forcing Generalized descriptive set theory Sierpinski's onto mapping principle xbox Foundations transformations Successor of Singular Cardinal Nonspecial tree P-Ideal Dichotomy L-space Cardinal function approachability ideal Ramsey theory over partitions Entangled linear order OCA Rock n' Roll Partition Relations Microscopic Approach Mandelbrot set regressive Souslin tree Lipschitz reduction Singular Density specializable Souslin tree AIM forcing sap Subtle cardinal super-Souslin tree Iterated forcing Respecting tree Parameterized proxy principle Open Access Knaster Uniformly homogeneous weak Kurepa tree Aronszajn tree Erdos Cardinal Uniformization weak square C-sequence Distributive tree Ascending path Rado's conjecture Interval topology on trees Subtle tree property Chang's conjecture Precaliber tensor product graph free Souslin tree Knaster and friends Absoluteness Almost countably chromatic Axiom R strongly bounded groups Martin's Axiom Diamond-sharp Universal Sequences Analytic sets Hedetniemi's conjecture Prevalent singular cardinals Non-saturation Whitehead Problem Diamond Diamond for trees countably metacompact middle diamond Commutative cancellative semigroups Singular cofinality Ostaszewski square Club Guessing PFA(S)[S] Forcing with side conditions Dowker space Jonsson cardinal Ineffable cardinal diamond star Shelah's Strong Hypothesis unbounded function Selective Ultrafilter Subnormal ideal SNR Singular cardinals combinatorics Filter reflection Rainbow sets Fast club Postprocessing function 54G20 ccc club_AD very good scale Sakurai's Bell inequality Closed coloring Souslin Tree Large Cardinals nonmeager set Generalized Clubs Chromatic number Uniformly coherent Hindman's Theorem Hereditarily Lindelöf space Cohen real Amenable C-sequence Fodor-type reflection Forcing Axioms ZFC construction positive partition relation S-Space Kurepa Hypothesis Constructible Universe Subadditive square principles Fat stationary set Reflecting stationary set indecomposable filter Well-behaved magma higher Baire space Monotonically far Slim tree coloring number Intersection model b-scale
Author Archives: Assaf Rinot
Logic in Hungary, August 2005
These are the slides of a contributed talk given at the Logic in Hungary 2005 meeting (Budapest, 5–11 August 2005). Talk Title: On the consistency strength of the Milner-Sauer Conjecture Abstract: In their paper from 1981, after learning about Pouzet‘s theorem that any … Continue reading
Posted in Contributed Talks
Tagged Antichain, Shelah's Strong Hypothesis, Singular cofinality
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A topological reflection principle equivalent to Shelah’s strong hypothesis
Abstract: We notice that Shelah’s Strong Hypothesis (SSH) is equivalent to the following reflection principle: Suppose $\mathbb X$ is an (infinite) first-countable space whose density is a regular cardinal, $\kappa$. If every separable subspace of $\mathbb X$ is of cardinality at most … Continue reading
Posted in Compactness, Publications, Topology
Tagged 03E04, 03E65, 54G15, Open Access, Shelah's Strong Hypothesis
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Openly generated Boolean algebras and the Fodor-type reflection principle
Joint work with Sakaé Fuchino. Abstract: We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is $\aleph _2$-projective. Previously it was known that this … Continue reading
Posted in Compactness, Publications
Tagged 03E35, 03E55, 03E65, 03E75, 03G05, 06E05, Axiom R, Fodor-type reflection, free Boolean algebra, projective Boolean algebra, Shelah's Strong Hypothesis, stationary reflection
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The failure of diamond on a reflecting stationary set
Joint work with Moti Gitik. Abstract: It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal. This should be comapred with … Continue reading
A relative of the approachability ideal, diamond and non-saturation
Abstract: Let $\lambda$ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square^*_\lambda$ together with $2^\lambda=\lambda^+$ implies $\diamondsuit_S$ for every $S\subseteq\lambda^+$ that reflects stationarily often. In this paper, for a subset $S\subset\lambda^+$, a normal subideal of … Continue reading
On guessing generalized clubs at the successors of regulars
Abstract: Konig, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of an higher Souslin tree from the strong guessing principle. Complementary to the author’s work on the validity of diamond and non-saturation … Continue reading
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
Transforming rectangles into squares, with applications to strong colorings
Abstract: It is proved that every singular cardinal $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$. As a … Continue reading
Antichains in partially ordered sets of singular cofinality
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 main result of of this … Continue reading
Posted in Publications, Singular Cardinals Combinatorics
Tagged 03E04, 03E35, 06A07, Antichain, Poset, Singular cofinality
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The Ostaszewski square, and homogeneous Souslin trees
Abstract: Assume GCH and let $\lambda$ denote an uncountable cardinal. We prove that if $\square_\lambda$ holds, then this may be witnessed by a coherent sequence $\left\langle C_\alpha \mid \alpha<\lambda^+\right\rangle$ with the following remarkable guessing property: For every sequence $\langle A_i\mid i<\lambda\rangle$ … Continue reading
Posted in Publications, Souslin Hypothesis, Squares and Diamonds
Tagged 03E05, 03E35, Club Guessing, Fat stationary set, Ostaszewski square, Souslin Tree
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