Archives
Keywords
Small forcing incompactness Was Ulam right? Strongly compact cardinal club_AD Precaliber Universal Sequences Partition Relations Hindman's Theorem Almost Souslin indecomposable ultrafilter Sierpinski's onto mapping principle GMA Reflecting stationary set diamond star Cardinal Invariants Filter reflection Open Access Countryman line Absoluteness Generalized descriptive set theory Foundations Strongly Luzin set Ostaszewski square Reduced Power PFA(S)[S] unbounded function Commutative cancellative semigroups coloring number Postprocessing function Aronszajn tree tensor product graph Uniformly coherent OCA Poset Knaster strongly bounded groups super-Souslin tree Parameterized proxy principle Subtle cardinal sap Analytic sets Chang's conjecture Amenable C-sequence Sigma-Prikry HOD Weakly compact cardinal transformations positive partition relation Shelah's Strong Hypothesis Axiom R Singular cofinality Intersection model AIM forcing Martin's Axiom SNR Diamond Subadditive Ulam matrix Almost countably chromatic ZFC construction Sakurai's Bell inequality Rainbow sets regressive Souslin tree Nonspecial tree Ineffable cardinal b-scale Dowker space Fat stationary set Closed coloring Commutative projection system Fast club Prevalent singular cardinals Square-Brackets Partition Relations Greatly Mahlo Successor of Regular Cardinal projective Boolean algebra Whitehead Problem 54G20 Dushnik-Miller very good scale xbox Distributive tree Selective Ultrafilter specializable Souslin tree Hereditarily Lindelöf space Mandelbrot set Minimal Walks Diamond-sharp ccc Subnormal ideal Knaster and friends Antichain countably metacompact Subtle tree property Luzin set P-Ideal Dichotomy PFA polarized partition relation Slim tree Kurepa Hypothesis S-Space free Boolean algebra Successor of Singular Cardinal Well-behaved magma Hedetniemi's conjecture free Souslin tree stationary hitting nonmeager set O-space square Club Guessing square principles Coherent tree weak diamond Singular cardinals combinatorics Fodor-type reflection L-space higher Baire space Large Cardinals Constructible Universe Local Club Condensation. Almost-disjoint family Iterated forcing Non-saturation Prikry-type forcing Jonsson cardinal Strong coloring weak Kurepa tree Vanishing levels Cohen real Ramsey theory over partitions stick Erdos-Hajnal graphs Souslin Tree full tree Uniformly homogeneous Rock n' Roll Generalized Clubs Respecting tree Rado's conjecture Cardinal function Ascent Path Lipschitz reduction approachability ideal Singular Density Forcing Axioms Erdos Cardinal Uniformization Forcing stationary reflection Chromatic number middle diamond reflection principles Microscopic Approach weak square C-sequence Diamond for trees
Category Archives: Publications
On topological spaces of singular density and minimal weight
Abstract: We introduce a weakening of the Generalized Continuum Hypothesis, which we will refer to as the Prevalent Singular cardinals Hypothesis (PSH), and show it implies that every topological space of density and weight $\aleph_{\omega_1}$ is not hereditarily Lindelöf. The assumption … Continue reading
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
Leave a comment
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
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
Leave a comment
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
5 Comments
