Archives
Keywords
Chang's conjecture positive partition relation 54G20 incompactness Small forcing Fodor-type reflection Successor of Regular Cardinal Sakurai's Bell inequality Strongly compact cardinal Reduced Power diamond star Intersection model Singular cardinals combinatorics Non-saturation Chromatic number Successor of Singular Cardinal Subtle cardinal club_AD sap Slim tree Aronszajn tree Subadditive Shelah's Strong Hypothesis Cohen real Iterated forcing transformations Coherent tree Entangled linear order regressive Souslin tree O-space Universal Sequences weak diamond S-Space Nonspecial tree Minimal Walks Absoluteness Cardinal function Subtle tree property square Rado's conjecture b-scale Mandelbrot set Ulam matrix strongly bounded groups xbox Knaster and friends Hedetniemi's conjecture tensor product graph Monotonically far Whitehead Problem Lipschitz reduction Ascent Path Greatly Mahlo Ineffable cardinal Foundations Weakly compact cardinal Microscopic Approach AIM forcing C-sequence PFA countably metacompact Fat stationary set indecomposable filter Selective Ultrafilter Dowker space Jonsson cardinal reflection principles Was Ulam right? coloring number Prevalent singular cardinals OCA perfectly normal Almost countably chromatic Distributive tree Partition Relations unbounded function square principles Ramsey theory over partitions Sierpinski's onto mapping principle Forcing with side conditions Postprocessing function Uniformly coherent Strong coloring Forcing Respecting tree Ascending path super-Souslin tree Sigma-Prikry Prikry-type forcing Almost Souslin Vanishing levels ZFC construction Fast club Kurepa Hypothesis very good scale Hereditarily Lindelöf space Open Access Interval topology on trees Reflecting stationary set Commutative projection system free Souslin tree nonmeager set Erdos Cardinal Dushnik-Miller Filter reflection Countryman line Local Club Condensation. Forcing Axioms projective Boolean algebra Strongly Luzin set Commutative cancellative semigroups Constructible Universe Club Guessing Almost-disjoint family Closed coloring Luzin set Singular Density Singular cofinality Parameterized proxy principle Poset SNR Well-behaved magma Analytic sets Precaliber stationary hitting Large Cardinals polarized partition relation stationary reflection Partition relations for trees weak Kurepa tree ccc approachability ideal Diamond Souslin Tree Uniformization Square-Brackets Partition Relations GMA Knaster HOD Antichain Generalized Clubs Martin's Axiom Subnormal ideal Amenable C-sequence free Boolean algebra Erdos-Hajnal graphs Diamond-sharp PFA(S)[S] P-Ideal Dichotomy specializable Souslin tree Rock n' Roll Generalized descriptive set theory weak square middle diamond stick Axiom R Diamond for trees Cardinal Invariants Ostaszewski square higher Baire space L-space Rainbow sets Hindman's Theorem Uniformly homogeneous full tree
Category Archives: Publications
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
