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