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