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free Boolean algebra Jonsson cardinal xbox Forcing Filter reflection Coherent tree Ascending path Ostaszewski square Whitehead Problem S-Space Rado's conjecture Almost-disjoint family Was Ulam right? Poset Uniformly homogeneous Dushnik-Miller C-sequence Successor of Regular Cardinal Distributive tree nonmeager set ccc Knaster Non-saturation Kurepa Hypothesis HOD weak square Shelah's Strong Hypothesis Strongly Luzin set Cardinal function Commutative projection system Generalized descriptive set theory Ramsey theory over partitions Subtle tree property super-Souslin tree Aronszajn tree Constructible Universe specializable Souslin tree Dowker space Singular cardinals combinatorics Hereditarily Lindelöf space Diamond-sharp Uniformization stick Subtle cardinal regressive Souslin tree tensor product graph AIM forcing Entangled linear order polarized partition relation PFA Sakurai's Bell inequality Cohen real Reflecting stationary set Axiom R Mandelbrot set Vanishing levels PFA(S)[S] Partition Relations Small forcing Closed coloring Fat stationary set Precaliber reflection principles Rock n' Roll L-space ZFC construction Amenable C-sequence Singular Density Selective Ultrafilter Minimal Walks indecomposable filter square principles Weakly compact cardinal approachability ideal incompactness Uniformly coherent Ineffable cardinal Microscopic Approach Diamond transformations Intersection model Fast club weak diamond Greatly Mahlo Absoluteness projective Boolean algebra Reduced Power Countryman line Almost countably chromatic diamond star Square-Brackets Partition Relations O-space Ascent Path club_AD Foundations Singular cofinality Subadditive P-Ideal Dichotomy Commutative cancellative semigroups Analytic sets Local Club Condensation. Antichain very good scale Luzin set Prevalent singular cardinals positive partition relation Iterated forcing Lipschitz reduction Fodor-type reflection Club Guessing Interval topology on trees Knaster and friends SNR coloring number Erdos-Hajnal graphs Diamond for trees Sierpinski's onto mapping principle Large Cardinals strongly bounded groups Partition relations for trees Chang's conjecture higher Baire space stationary hitting Strongly compact cardinal OCA Souslin Tree Strong coloring Erdos Cardinal full tree Postprocessing function Ulam matrix unbounded function Forcing Axioms free Souslin tree Martin's Axiom Well-behaved magma Almost Souslin GMA middle diamond sap Sigma-Prikry Universal Sequences 54G20 stationary reflection Open Access Rainbow sets weak Kurepa tree countably metacompact Subnormal ideal Hindman's Theorem square Cardinal Invariants Hedetniemi's conjecture Chromatic number b-scale Slim tree Prikry-type forcing Parameterized proxy principle Monotonically far Forcing with side conditions Nonspecial tree Generalized Clubs Respecting tree perfectly normal Successor of Singular Cardinal
Tag Archives: Square-Brackets Partition Relations
Sums of triples in Abelian groups
Joint work with Ido Feldman. Abstract. Motivated by a problem in additive Ramsey theory, we extend Todorcevic’s partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for … 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
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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
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
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Strong failures of higher analogs of Hindman’s Theorem
Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading
Posted in Groups, Partition Relations, Publications
Tagged 03E02, 03E35, 03E75, 05A17, 05D10, 11P99, 20M14, Chang's conjecture, Commutative cancellative semigroups, Entangled linear order, Erdos Cardinal, Hindman's Theorem, Jonsson cardinal, Kurepa Hypothesis, Square-Brackets Partition Relations, Weakly compact cardinal, ZFC construction
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Prolific Souslin trees
In a paper from 1971, Erdos and Hajnal asked whether (assuming CH) every coloring witnessing $\aleph_1\nrightarrow[\aleph_1]^2_3$ has a rainbow triangle. The negative solution was given in a 1975 paper by Shelah, and the proof and relevant definitions may be found … Continue reading
Posted in Blog, Expository
Tagged Rainbow sets, Souslin Tree, Square-Brackets Partition Relations
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Complicated colorings
Abstract. If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^\lambda_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. (Recall that $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ asserts the existence of a coloring $d:[\lambda]^2\rightarrow\lambda$ such that for any family $\mathcal A\subseteq[\lambda]^{<\kappa}$ of size $\lambda$, consisting of pairwise … Continue reading
Posted in Partition Relations, Publications
Tagged Minimal Walks, Open Access, Square-Brackets Partition Relations
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MFO workshop in Set Theory, January 2014
I gave an invited talk at the Set Theory workshop in Obwerwolfach, January 2014. Talk Title: Complicated Colorings. Abstract: If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^{\lambda}_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. Downloads:
Rectangular square-bracket operation for successor of regular cardinals
Joint work with Stevo Todorcevic. Extended Abstract: Consider the coloring statement $\lambda^+\nrightarrow[\lambda^+;\lambda^+]^2_{\lambda^+}$ for a given regular cardinal $\lambda$: In 1990, Shelah proved the above for $\lambda>2^{\aleph_0}$; In 1991, Shelah proved the above for $\lambda>\aleph_1$; In 1997, Shelah proved the above … Continue reading
Comparing rectangles with squares through rainbow sets
In Todorcevic’s class last week, he proved all the results of Chapter 8 from his Walks on Ordinals book, up to (and including) Theorem 8.1.11. The upshots are as follows: Every regular infinite cardinal $\theta$ admits a naturally defined function … Continue reading
