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