Archives
Keywords
Local Club Condensation. perfectly normal stick Open Access Respecting tree Whitehead Problem Diamond-sharp Ascending path Club Guessing Universal Sequences Iterated forcing SNR Luzin set Knaster coloring number Ulam matrix Antichain Large Cardinals Subnormal ideal Analytic sets Subtle cardinal stationary hitting Forcing with side conditions Entangled linear order middle diamond 54G20 Sigma-Prikry xbox P-Ideal Dichotomy Fast club Forcing Axioms Shelah's Strong Hypothesis Almost-disjoint family Generalized descriptive set theory club_AD super-Souslin tree Small forcing Singular cardinals combinatorics very good scale Microscopic Approach Hedetniemi's conjecture higher Baire space Uniformly coherent Cardinal function nonmeager set approachability ideal Knaster and friends Cardinal Invariants b-scale indecomposable filter regressive Souslin tree Postprocessing function countably metacompact stationary reflection Diamond for trees free Souslin tree O-space OCA specializable Souslin tree Forcing HOD Commutative projection system Uniformly homogeneous Vanishing levels ccc C-sequence full tree Closed coloring Rock n' Roll sap Filter reflection Kurepa Hypothesis Rainbow sets Lipschitz reduction weak square Erdos Cardinal Interval topology on trees Hereditarily Lindelöf space S-Space Reduced Power Square-Brackets Partition Relations ZFC construction Intersection model Non-saturation Parameterized proxy principle Dushnik-Miller PFA Martin's Axiom Successor of Regular Cardinal Nonspecial tree Almost countably chromatic Prikry-type forcing Countryman line L-space Fat stationary set Singular cofinality Subtle tree property reflection principles Subadditive Singular Density Strongly Luzin set Strong coloring square principles positive partition relation Well-behaved magma Commutative cancellative semigroups transformations Ramsey theory over partitions Rado's conjecture Chang's conjecture strongly bounded groups free Boolean algebra incompactness Minimal Walks Jonsson cardinal Axiom R Precaliber Monotonically far Hindman's Theorem Ineffable cardinal Ascent Path Mandelbrot set Coherent tree square Amenable C-sequence polarized partition relation Distributive tree Dowker space Diamond Greatly Mahlo Sakurai's Bell inequality Partition Relations unbounded function Ostaszewski square weak diamond diamond star Foundations Absoluteness Partition relations for trees Generalized Clubs Was Ulam right? GMA Strongly compact cardinal Fodor-type reflection Erdos-Hajnal graphs projective Boolean algebra Slim tree weak Kurepa tree Successor of Singular Cardinal tensor product graph Constructible Universe Sierpinski's onto mapping principle Weakly compact cardinal Cohen real AIM forcing Aronszajn tree Prevalent singular cardinals Poset Souslin Tree PFA(S)[S] Uniformization Chromatic number Reflecting stationary set Selective Ultrafilter Almost Souslin
Tag Archives: Minimal Walks
Was Ulam right? III: Indecomposable ideals
Joint work with Tanmay Inamdar. Abstract. We continue our study of Ulam’s measure problem. In contrast to our previous works, we shift our focus from measures stratified by their additivity, to measures stratified by their indecomposability. The breakthrough here is … Continue reading
Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines
Joint work with Tanmay Inamdar. Abstract. We investigate global structural properties of linear orders of a fixed infinite size. It is classical that the countable linear orders and the continuum-sized orders exhibit contrasting behaviours. Modern results show that strong extensions … Continue reading
Posted in Basis problems, Partition Relations, Preprints
Tagged Aronszajn tree, Ascending path, Club Guessing, Countryman line, Entangled linear order, Minimal Walks, Monotonically far, Partition relations for trees, Strong coloring, Subtle tree property, Vanishing levels, ZFC construction
2 Comments
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
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
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
6th European Set Theory Conference, July 2017
I gave a 3-lecture tutorial at the 6th European Set Theory Conference in Budapest, July 2017. Title: Strong colorings and their applications. Abstract. Consider the following questions. Is the product of two $\kappa$-cc partial orders again $\kappa$-cc? Does there exist … Continue reading
Posted in Invited Talks, Open Problems
Tagged b-scale, Cohen real, Luzin set, Minimal Walks, Souslin Tree, Square-Brackets Partition Relations
4 Comments
Distributive Aronszajn trees
Joint work with Ari Meir Brodsky. Abstract. Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here, it is proved that the same conclusion remains … Continue reading
Square with built-in diamond-plus
Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, Respecting tree, square, xbox
1 Comment
Chain conditions of products, and weakly compact cardinals
Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading
Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
2 Comments
