Archives
Keywords
club_AD incompactness Sierpinski's onto mapping principle Reflecting stationary set Hindman's Theorem Singular cofinality ccc Chang's conjecture Knaster and friends Foundations Cohen real Dushnik-Miller Martin's Axiom Interval topology on trees Whitehead Problem Selective Ultrafilter Fodor-type reflection Aronszajn tree Successor of Singular Cardinal HOD GMA Small forcing sap Precaliber Mandelbrot set regressive Souslin tree Amenable C-sequence Commutative cancellative semigroups Ineffable cardinal Subadditive Singular cardinals combinatorics specializable Souslin tree Ascending path PFA square Subtle cardinal P-Ideal Dichotomy Respecting tree Large Cardinals Countryman line middle diamond Hereditarily Lindelöf space Strongly compact cardinal Vanishing levels Reduced Power Axiom R Nonspecial tree Cardinal Invariants nonmeager set Greatly Mahlo Fast club Square-Brackets Partition Relations higher Baire space stick Rado's conjecture ZFC construction strongly bounded groups Iterated forcing diamond star Distributive tree perfectly normal PFA(S)[S] weak square reflection principles Successor of Regular Cardinal Club Guessing Weakly compact cardinal Partition Relations weak Kurepa tree C-sequence Singular Density Antichain SNR super-Souslin tree Diamond for trees Almost countably chromatic Rainbow sets stationary hitting Kurepa Hypothesis Dowker space Almost-disjoint family free Boolean algebra Prikry-type forcing indecomposable filter Almost Souslin Ascent Path Generalized Clubs very good scale Chromatic number Luzin set projective Boolean algebra Ulam matrix Prevalent singular cardinals Postprocessing function b-scale Uniformly coherent L-space Slim tree Universal Sequences Intersection model Non-saturation Subnormal ideal Filter reflection Uniformization Diamond tensor product graph O-space Absoluteness Minimal Walks Shelah's Strong Hypothesis Rock n' Roll Fat stationary set Strong coloring Well-behaved magma Sakurai's Bell inequality Ramsey theory over partitions Forcing with side conditions Poset Forcing Entangled linear order Analytic sets 54G20 weak diamond OCA Cardinal function Lipschitz reduction Knaster Sigma-Prikry Was Ulam right? Ostaszewski square Souslin Tree polarized partition relation transformations Strongly Luzin set Jonsson cardinal Commutative projection system Forcing Axioms Parameterized proxy principle Closed coloring Partition relations for trees S-Space coloring number stationary reflection Local Club Condensation. positive partition relation Uniformly homogeneous unbounded function Open Access Diamond-sharp Microscopic Approach Coherent tree Monotonically far Erdos-Hajnal graphs Generalized descriptive set theory free Souslin tree Hedetniemi's conjecture approachability ideal Constructible Universe Erdos Cardinal AIM forcing countably metacompact xbox full tree Subtle tree property square principles
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 $R_2$
