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