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