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