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Ramsey theory over partitions Local Club Condensation. diamond star Minimal Walks countably metacompact Square-Brackets Partition Relations Cardinal Invariants Luzin set Partition relations for trees Rainbow sets Lipschitz reduction Nonspecial tree Vanishing levels positive partition relation transformations Kurepa Hypothesis indecomposable filter Successor of Singular Cardinal Strongly compact cardinal Successor of Regular Cardinal C-sequence Diamond-sharp Almost countably chromatic OCA Singular cardinals combinatorics Knaster and friends Rock n' Roll Sakurai's Bell inequality Shelah's Strong Hypothesis AIM forcing Universal Sequences Small forcing Singular Density Greatly Mahlo Generalized Clubs xbox Strong coloring middle diamond Interval topology on trees Prikry-type forcing Commutative cancellative semigroups Chromatic number Slim tree stick Entangled linear order Was Ulam right? perfectly normal HOD ccc Diamond for trees Hereditarily Lindelöf space Souslin Tree S-Space nonmeager set L-space Almost Souslin sap projective Boolean algebra strongly bounded groups Sierpinski's onto mapping principle Mandelbrot set Countryman line free Boolean algebra Subtle cardinal Absoluteness Diamond regressive Souslin tree free Souslin tree Axiom R weak Kurepa tree Uniformization GMA Dushnik-Miller Forcing Jonsson cardinal incompactness Fat stationary set weak diamond Almost-disjoint family Cardinal function full tree Constructible Universe Fodor-type reflection Ineffable cardinal Distributive tree Coherent tree Hindman's Theorem Monotonically far Respecting tree polarized partition relation club_AD square principles Ascent Path Strongly Luzin set specializable Souslin tree Intersection model PFA Open Access Dowker space super-Souslin tree Ascending path Singular cofinality very good scale Large Cardinals Amenable C-sequence Poset square weak square Generalized descriptive set theory Erdos-Hajnal graphs Ulam matrix unbounded function O-space Microscopic Approach Postprocessing function Filter reflection Erdos Cardinal Forcing with side conditions Aronszajn tree Antichain coloring number ZFC construction Selective Ultrafilter approachability ideal Uniformly homogeneous Knaster Prevalent singular cardinals stationary reflection Reduced Power Rado's conjecture Subnormal ideal stationary hitting Analytic sets Foundations Martin's Axiom Commutative projection system Precaliber Parameterized proxy principle tensor product graph PFA(S)[S] Subadditive Whitehead Problem Reflecting stationary set higher Baire space Sigma-Prikry reflection principles Iterated forcing Weakly compact cardinal 54G20 Subtle tree property Well-behaved magma P-Ideal Dichotomy Non-saturation Cohen real Hedetniemi's conjecture Uniformly coherent b-scale Chang's conjecture Partition Relations Forcing Axioms Closed coloring Ostaszewski square Club Guessing Fast club SNR
Category Archives: Publications
A microscopic approach to Souslin-tree constructions. Part I
Joint work with Ari Meir Brodsky. Abstract. We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E65, 05C05, Coherent tree, Diamond, Microscopic Approach, Parameterized proxy principle, Slim tree, Souslin Tree, square, xbox
5 Comments
Square with built-in diamond-plus
Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, Respecting tree, square, xbox
1 Comment
Reduced powers of Souslin trees
Joint work with Ari Meir Brodsky. Abstract. We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed … Continue reading
Putting a diamond inside the square
Abstract. By a 35-year-old theorem of Shelah, $\square_\lambda+\diamondsuit(\lambda^+)$ does not imply square-with-built-in-diamond_lambda for regular uncountable cardinals $\lambda$. Here, it is proved that $\square_\lambda+\diamondsuit(\lambda^+)$ is equivalent to square-with-built-in-diamond_lambda for every singular cardinal $\lambda$. Downloads: Citation information: A. Rinot, Putting a diamond inside … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Diamond, square, Successor of Singular Cardinal
1 Comment
Same Graph, Different Universe
Abstract. May the same graph admit two different chromatic numbers in two different universes? how about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Godel’s constructible … Continue reading
Posted in Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, approachability ideal, Chromatic number, Constructible Universe, Forcing, Ostaszewski square
10 Comments
Chain conditions of products, and weakly compact cardinals
Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading
Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
2 Comments
Complicated colorings
Abstract. If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^\lambda_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. (Recall that $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ asserts the existence of a coloring $d:[\lambda]^2\rightarrow\lambda$ such that for any family $\mathcal A\subseteq[\lambda]^{<\kappa}$ of size $\lambda$, consisting of pairwise … Continue reading
Posted in Partition Relations, Publications
Tagged Minimal Walks, Open Access, Square-Brackets Partition Relations
2 Comments
Hedetniemi’s conjecture for uncountable graphs
Abstract. It is proved that in Godel’s constructible universe, for every successor cardinal $\kappa$, there exist graphs $\mathcal G$ and $\mathcal H$ of size and chromatic number $\kappa$, for which the tensor product graph $\mathcal G\times\mathcal H$ is countably chromatic. … Continue reading
Chromatic numbers of graphs – large gaps
Abstract. We say that a graph $G$ is $(\aleph_0,\kappa)$-chromatic if $\text{Chr}(G)=\kappa$, while $\text{Chr}(G’)\le\aleph_0$ for any subgraph $G’$ of $G$ of size $<|G|$. The main result of this paper reads as follows. If $\square_\lambda+\text{CH}_\lambda$ holds for a given uncountable cardinal $\lambda$, … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Almost countably chromatic, Chromatic number, incompactness, Ostaszewski square
6 Comments
Rectangular square-bracket operation for successor of regular cardinals
Joint work with Stevo Todorcevic. Extended Abstract: Consider the coloring statement $\lambda^+\nrightarrow[\lambda^+;\lambda^+]^2_{\lambda^+}$ for a given regular cardinal $\lambda$: In 1990, Shelah proved the above for $\lambda>2^{\aleph_0}$; In 1991, Shelah proved the above for $\lambda>\aleph_1$; In 1997, Shelah proved the above … Continue reading
