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Vanishing levels Whitehead Problem Strongly compact cardinal Antichain Ascending path Fast club reflection principles Subadditive club_AD Intersection model Souslin Tree Forcing with side conditions Successor of Regular Cardinal Cardinal Invariants Diamond-sharp Hedetniemi's conjecture Poset GMA stationary reflection HOD Cohen real Axiom R Diamond for trees b-scale Partition relations for trees Mandelbrot set higher Baire space Universal Sequences Uniformization Partition Relations Rock n' Roll stationary hitting Strongly Luzin set Large Cardinals Knaster and friends Absoluteness Dowker space Luzin set Non-saturation Lipschitz reduction Ostaszewski square Hindman's Theorem Interval topology on trees weak Kurepa tree Almost countably chromatic xbox Jonsson cardinal Singular cofinality diamond star polarized partition relation middle diamond Generalized Clubs Nonspecial tree Subtle cardinal projective Boolean algebra full tree Aronszajn tree Iterated forcing Rainbow sets Open Access Ascent Path square weak diamond Dushnik-Miller sap Rado's conjecture Uniformly homogeneous coloring number Square-Brackets Partition Relations AIM forcing perfectly normal Ulam matrix unbounded function Amenable C-sequence OCA strongly bounded groups Reflecting stationary set Commutative projection system Chromatic number incompactness indecomposable filter Greatly Mahlo regressive Souslin tree Sierpinski's onto mapping principle Postprocessing function tensor product graph ZFC construction Closed coloring Erdos Cardinal Sakurai's Bell inequality Ramsey theory over partitions free Boolean algebra C-sequence Respecting tree Analytic sets Commutative cancellative semigroups Parameterized proxy principle O-space Forcing Axioms Monotonically far Weakly compact cardinal specializable Souslin tree Diamond Erdos-Hajnal graphs ccc Sigma-Prikry Local Club Condensation. Knaster Well-behaved magma Forcing Singular cardinals combinatorics Kurepa Hypothesis Slim tree Reduced Power Generalized descriptive set theory positive partition relation square principles PFA(S)[S] Filter reflection Countryman line Foundations nonmeager set L-space Microscopic Approach free Souslin tree Was Ulam right? Almost Souslin Cardinal function Fodor-type reflection Hereditarily Lindelöf space Ineffable cardinal Club Guessing Subnormal ideal Selective Ultrafilter super-Souslin tree Strong coloring Prikry-type forcing approachability ideal Precaliber PFA Shelah's Strong Hypothesis SNR very good scale Coherent tree Singular Density Almost-disjoint family stick Minimal Walks 54G20 Entangled linear order Martin's Axiom weak square Uniformly coherent Small forcing Chang's conjecture countably metacompact P-Ideal Dichotomy Constructible Universe transformations Fat stationary set Distributive tree Subtle tree property Prevalent singular cardinals S-Space Successor of Singular Cardinal
Author Archives: Assaf Rinot
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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RIMS workshop on Set Theory 2025
I gave an online contributed talk at the RIMS workshop on Set Theory 2025 in Kyoto, December 2025. Talk Title: What is a higher forcing axiom? Abstract: There are multiple interpretations of what is a forcing axiom. We shall survey … Continue reading
The 18th International Workshop on Set Theory in Luminy, November 2015
I gave an invited talk at the 18th International Workshop on Set Theory in Luminy in Marseille, November 2025. Talk Title: What is a higher forcing axiom? Abstract: There are multiple interpretations of what is a forcing axiom. We shall … Continue reading
Posted in Invited Talks
Tagged Forcing Axioms
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Conference in honor of Saharon Shelah’s 80th birthday
I gave an invited talk at the conference in honor of Shelah’s 80th birthday, July 2025. Talk Title: Marginalia to [Sh:365] Abstract: Solovay famously proved that every stationary subset of a regular uncountable cardinal kappa may be decomposed into kappa … Continue reading
Posted in Invited Talks
Tagged Greatly Mahlo, Strong coloring, Was Ulam right?
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2025 Annual conference of the IMU
I gave an invited talk at the Set Theory session of the annual meeting of the IMU, July 2025. Talk Title: What my co-authors taught me about indecomposability Abstract: Ulam’s measure problem studies when there is a countably-additive two-valued measure … Continue reading
Posted in Invited Talks
Tagged indecomposable filter
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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
A new model for all C-sequences are trivial
Joint work with Zhixing You and Jiachen Yuan. Abstract. We construct a model in which all C-sequences are trivial, yet there exists a $\kappa$-Souslin tree with full vanishing levels. This answers a question from a previous paper, and provides an … Continue reading
Posted in Compactness, Publications
Tagged Ascent Path, C-sequence, Intersection model, Souslin Tree, Subtle tree property, Vanishing levels
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MFO workshop in Set Theory, January 2025
I gave an invited talk at the Set Theory meeting in Obwerwolfach, January 2025. Talk Title: Non-structure theorems for higher Aronszajn lines. Abstract: An $\omega_1$-Countryman line is an uncountable linear order $L$ such that $L^2$ (with the pointwise ordering) is … Continue reading
Posted in Invited Talks
Tagged Countryman line, Monotonically far
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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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