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