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Prikry-type forcing Erdos-Hajnal graphs Non-saturation Diamond-sharp Generalized descriptive set theory Kurepa Hypothesis PFA free Boolean algebra Well-behaved magma Subnormal ideal Dowker space Entangled linear order Ascent Path perfectly normal coloring number free Souslin tree approachability ideal Sigma-Prikry Martin's Axiom Cardinal function Dushnik-Miller Filter reflection Shelah's Strong Hypothesis P-Ideal Dichotomy Aronszajn tree Selective Ultrafilter Singular cardinals combinatorics 54G20 Almost Souslin Fodor-type reflection transformations Was Ulam right? Microscopic Approach Antichain square Amenable C-sequence higher Baire space Constructible Universe Diamond Subadditive Vanishing levels Poset ZFC construction specializable Souslin tree Commutative cancellative semigroups indecomposable filter Whitehead Problem Almost countably chromatic ccc tensor product graph very good scale Souslin Tree Chromatic number C-sequence Postprocessing function Universal Sequences incompactness Uniformization club_AD Axiom R Closed coloring GMA Partition Relations strongly bounded groups unbounded function Open Access SNR middle diamond Strong coloring polarized partition relation weak diamond diamond star Hindman's Theorem Weakly compact cardinal Singular Density Chang's conjecture nonmeager set L-space Minimal Walks O-space Parameterized proxy principle Forcing Coherent tree Subtle tree property S-Space full tree xbox Rado's conjecture Fat stationary set Interval topology on trees Partition relations for trees Forcing with side conditions Lipschitz reduction regressive Souslin tree Knaster and friends Sakurai's Bell inequality Generalized Clubs Mandelbrot set Diamond for trees Iterated forcing Analytic sets Intersection model Club Guessing Reduced Power Subtle cardinal Cohen real weak Kurepa tree Strongly Luzin set Hereditarily Lindelöf space super-Souslin tree weak square Luzin set reflection principles stationary reflection Monotonically far Square-Brackets Partition Relations AIM forcing Countryman line Sierpinski's onto mapping principle b-scale Successor of Regular Cardinal Ascending path Small forcing Precaliber Knaster Ineffable cardinal Large Cardinals Uniformly homogeneous Hedetniemi's conjecture stick Reflecting stationary set Greatly Mahlo Local Club Condensation. Distributive tree HOD Rock n' Roll Erdos Cardinal Uniformly coherent Slim tree Respecting tree Cardinal Invariants Ramsey theory over partitions PFA(S)[S] Ulam matrix positive partition relation Ostaszewski square Almost-disjoint family Jonsson cardinal Prevalent singular cardinals Nonspecial tree Strongly compact cardinal Successor of Singular Cardinal projective Boolean algebra Rainbow sets Singular cofinality square principles stationary hitting Absoluteness countably metacompact Foundations OCA Commutative projection system sap Fast club Forcing Axioms
Author Archives: Assaf Rinot
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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More notions of forcing add a square
Joint work with Yair Hayut and Zhixing You. Abstract. Foreman and Magidor showed that the continuum hypothesis implies the existence of a countably-closed $\aleph_2$-cc forcing notion $\mathbb P$ for adding $\square_{\omega_1}$. Here, we show that $\mathbb P$ may consistently be … Continue reading
Posted in Squares and Diamonds
Tagged full tree, Souslin Tree, square principles, super-Souslin tree
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40th summer conference on topology and its applications
I gave an invited talk at the General and Set-Theoretic Topology session of the 40th summer conference on topology and its applications, July 2026. Talk title: A tree-based perfectly normal space whose square is not countably metacompact. Abstract. The study … Continue reading
Posted in Invited Talks, Topology
Tagged Interval topology on trees
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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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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
