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nonmeager set Closed coloring positive partition relation Shelah's Strong Hypothesis Well-behaved magma Whitehead Problem Strongly compact cardinal Non-saturation stick Aronszajn tree Selective Ultrafilter square Diamond for trees Monotonically far projective Boolean algebra Generalized Clubs square principles Successor of Singular Cardinal Rado's conjecture Countryman line Weakly compact cardinal Forcing Axioms Dowker space Strongly Luzin set Large Cardinals Erdos Cardinal Almost Souslin HOD PFA Almost countably chromatic Sigma-Prikry Kurepa Hypothesis S-Space Entangled linear order Reduced Power Commutative cancellative semigroups very good scale Precaliber approachability ideal strongly bounded groups Ulam matrix Almost-disjoint family coloring number indecomposable filter Uniformly homogeneous PFA(S)[S] Commutative projection system sap Rainbow sets unbounded function C-sequence Club Guessing ccc O-space Prevalent singular cardinals Singular cofinality Ostaszewski square Martin's Axiom Prikry-type forcing Partition Relations polarized partition relation Axiom R diamond star Iterated forcing perfectly normal Mandelbrot set Constructible Universe Forcing Poset Ramsey theory over partitions Uniformization Distributive tree Subtle tree property AIM forcing Ascending path weak Kurepa tree Forcing with side conditions Erdos-Hajnal graphs Slim tree stationary reflection Successor of Regular Cardinal Local Club Condensation. P-Ideal Dichotomy Hedetniemi's conjecture Respecting tree weak diamond middle diamond free Souslin tree Hereditarily Lindelöf space Greatly Mahlo Chang's conjecture Vanishing levels b-scale specializable Souslin tree Open Access Interval topology on trees Antichain Minimal Walks L-space Square-Brackets Partition Relations Ineffable cardinal countably metacompact xbox higher Baire space Nonspecial tree Small forcing regressive Souslin tree Filter reflection Diamond-sharp transformations Singular cardinals combinatorics Dushnik-Miller Fodor-type reflection Parameterized proxy principle Cohen real Cardinal function super-Souslin tree Coherent tree Amenable C-sequence Cardinal Invariants SNR Subnormal ideal Subtle cardinal Ascent Path Foundations Subadditive Knaster Rock n' Roll Jonsson cardinal Souslin Tree full tree Singular Density 54G20 Postprocessing function ZFC construction Knaster and friends Uniformly coherent Hindman's Theorem Partition relations for trees Absoluteness Was Ulam right? reflection principles Lipschitz reduction free Boolean algebra Universal Sequences Microscopic Approach Intersection model GMA Chromatic number OCA incompactness Sakurai's Bell inequality Strong coloring Luzin set Generalized descriptive set theory Fat stationary set club_AD Diamond Fast club Sierpinski's onto mapping principle weak square tensor product graph Analytic sets stationary hitting Reflecting stationary set
Category Archives: Partition Relations
Partition relations for trees II: Entangled linear orders
Joint work with Tanmay Inamdar. Abstract. We find new sufficient conditions for the existence of entangled linear orders. The constructions use anti-Ramsey colourings for trees, and they provide a fine control on the extent of entangledness. As an application, from … Continue reading
Posted in Partition Relations, Preprints
Tagged Aronszajn tree, Entangled linear order, Partition relations for trees, Strongly Luzin set
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Partition relations for trees I: Incomparable trees
Joint work with Tanmay Inamdar. Abstract. Todorcevic proved that Martin’s axiom implies that every two coherent $\aleph_1$-Aronszajn trees are comparable. Here, from cardinal arithmetic assumptions, we obtain the failure of the analogous statement for higher trees. In particular, for every … Continue reading
Posted in Partition Relations, Publications
Tagged Lipschitz reduction, Partition relations for trees
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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
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A counterexample related to a theorem of Komjáth and Weiss
Joint work with Rodrigo Rey Carvalho. Abstract. In a paper from 1987, Komjath and Weiss proved that for every regular topological space $X$ of character less than $\mathfrak b$, if $X\rightarrow(\text{top }{\omega+1})^1_\omega$, then $X\rightarrow(\text{top }{\alpha})^1_\omega$ for all $\alpha<\omega_1$. In addition, … Continue reading
Posted in Partition Relations, Preprints, Topology
Tagged 03E02, 54G20, Open Access, Prikry-type forcing, ZFC construction
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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
Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems
Joint work with Menachem Kojman and Juris Steprāns. Abstract. In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in … Continue reading
Was Ulam right? II: Small width and general ideals
Joint work with Tanmay Inamdar. Abstract. We continue our study of Sierpinski-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam’s theorem and its extension … Continue reading
Posted in Partition Relations, Publications
Tagged 03E02, 03E35, 03E55, C-sequence, Kurepa Hypothesis, Open Access, Subnormal ideal, Ulam matrix, Was Ulam right?
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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
Was Ulam right? I: Basic theory and subnormal ideals
Joint work with Tanmay Inamdar. Abstract. We introduce various coloring principles which generalize the so-called onto mapping principle of Sierpinski to larger cardinals and general ideals. We prove that these principles capture the notion of an Ulam matrix and allow … Continue reading
