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