Archives
Keywords
ccc Forcing with side conditions stick Subtle tree property Fat stationary set polarized partition relation Singular cofinality Uniformly coherent Dushnik-Miller O-space transformations Coherent tree SNR C-sequence Forcing Uniformly homogeneous indecomposable filter Kurepa Hypothesis Cardinal function Prevalent singular cardinals Foundations Sigma-Prikry Local Club Condensation. Strongly compact cardinal Monotonically far Ascent Path Successor of Regular Cardinal Successor of Singular Cardinal Chang's conjecture Almost countably chromatic Cardinal Invariants positive partition relation Ostaszewski square diamond star unbounded function Iterated forcing Parameterized proxy principle S-Space Sakurai's Bell inequality Rock n' Roll perfectly normal Rainbow sets Ramsey theory over partitions Erdos-Hajnal graphs Prikry-type forcing Poset PFA weak Kurepa tree Intersection model Martin's Axiom Shelah's Strong Hypothesis Reflecting stationary set sap coloring number Souslin Tree Constructible Universe approachability ideal Analytic sets Postprocessing function Ineffable cardinal Cohen real Amenable C-sequence Whitehead Problem Club Guessing Erdos Cardinal Hedetniemi's conjecture square Small forcing Dowker space Interval topology on trees Open Access Generalized descriptive set theory Minimal Walks reflection principles Generalized Clubs Microscopic Approach Weakly compact cardinal Almost Souslin full tree countably metacompact Subadditive Knaster Almost-disjoint family Ulam matrix super-Souslin tree Universal Sequences PFA(S)[S] very good scale Entangled linear order Mandelbrot set OCA Jonsson cardinal Selective Ultrafilter Rado's conjecture stationary reflection middle diamond Reduced Power stationary hitting Antichain Filter reflection Strongly Luzin set higher Baire space Slim tree Lipschitz reduction Nonspecial tree Subnormal ideal Countryman line nonmeager set L-space specializable Souslin tree Absoluteness Non-saturation Hindman's Theorem Distributive tree Commutative projection system Diamond for trees incompactness Uniformization Axiom R Singular Density club_AD Fodor-type reflection Was Ulam right? Well-behaved magma Strong coloring Singular cardinals combinatorics Partition relations for trees Closed coloring GMA free Boolean algebra Square-Brackets Partition Relations strongly bounded groups projective Boolean algebra AIM forcing Fast club Ascending path regressive Souslin tree Sierpinski's onto mapping principle P-Ideal Dichotomy Diamond-sharp Partition Relations Forcing Axioms tensor product graph 54G20 b-scale Aronszajn tree Greatly Mahlo Subtle cardinal free Souslin tree Large Cardinals Diamond weak square square principles xbox weak diamond Vanishing levels Respecting tree Hereditarily Lindelöf space Knaster and friends Chromatic number Luzin set ZFC construction Precaliber Commutative cancellative semigroups HOD
Tag Archives: square
Complicated colorings, revisited
Joint work with Jing Zhang. Abstract. In a paper from 1997, Shelah asked whether $Pr_1(\lambda^+,\lambda^+,\lambda^+,\lambda)$ holds for every inaccessible cardinal $\lambda$. Here, we prove that an affirmative answer follows from $\square(\lambda^+)$. Furthermore, we establish that for every pair $\chi<\kappa$ of … Continue reading
Knaster and friends III: Subadditive colorings
Joint work with Chris Lambie-Hanson. Abstract. We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta < \kappa$, the existence … Continue reading
Strongest transformations
Joint work with Jing Zhang. Abstract. We continue our study of maps transforming high-dimensional complicated objects into squares of stationary sets. Previously, we proved that many such transformations exist in ZFC, and here we address the consistency of the strongest … Continue reading
Posted in Partition Relations, Publications
Tagged Diamond, Minimal Walks, square, Square-Brackets Partition Relations, stick, transformations, xbox
2 Comments
Transformations of the transfinite plane
Joint work with Jing Zhang. Abstract. We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every … Continue reading
Knaster and friends II: The C-sequence number
Joint work with Chris Lambie-Hanson. Abstract. Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable … Continue reading
11th Young Set Theory Workshop, June 2018
I gave a 4-lecture tutorial at the 11th Young Set Theory Workshop, Lausanne, June 2018. Title: In praise of C-sequences. Abstract. Ulam and Solovay showed that any stationary set may be split into two. Is it also the case that … Continue reading
Posted in Invited Talks
Tagged Aronszajn tree, C-sequence, incompactness, Knaster, Minimal Walks, Postprocessing function, square
Leave a comment
Knaster and friends I: Closed colorings and precalibers
Joint work with Chris Lambie-Hanson. Abstract. The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $\kappa$-cc posets whose squares … Continue reading
A forcing axiom deciding the generalized Souslin Hypothesis
Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, Souslin Tree, square, super-Souslin tree
1 Comment
Reflection on the coloring and chromatic numbers
Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Chang's conjecture, Chromatic number, coloring number, Fodor-type reflection, incompactness, Iterated forcing, Parameterized proxy principle, Postprocessing function, Rado's conjecture, square, stationary reflection
2 Comments
The reflection principle $R_2$
A few years ago, in this paper, I introduced the following reflection principle: Definition. $R_2(\theta,\kappa)$ asserts that for every function $f:E^\theta_{<\kappa}\rightarrow\kappa$, there exists some $j<\kappa$ for which the following set is nonstationary: $$A_j:=\{\delta\in E^\theta_\kappa\mid f^{-1}[j]\cap\delta\text{ is nonstationary}\}.$$ I wrote there … Continue reading
Posted in Blog
Tagged reflection principles, square, stationary reflection, Weakly compact cardinal
Comments Off on The reflection principle $R_2$
