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