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