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