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weak diamond middle diamond Countryman line OCA Square-Brackets Partition Relations Diamond-sharp SNR Generalized Clubs Partition Relations square principles Forcing Axioms stick Iterated forcing Uniformly homogeneous ZFC construction Uniformization Prevalent singular cardinals Subnormal ideal weak Kurepa tree Diamond Sierpinski's onto mapping principle C-sequence Erdos Cardinal super-Souslin tree Entangled linear order Rado's conjecture Ascent Path HOD xbox Selective Ultrafilter Forcing Singular cofinality stationary reflection Respecting tree Ineffable cardinal Partition relations for trees Precaliber Commutative cancellative semigroups Knaster and friends Subtle tree property Lipschitz reduction Dowker space Large Cardinals Coherent tree Open Access Greatly Mahlo Uniformly coherent positive partition relation Dushnik-Miller L-space projective Boolean algebra stationary hitting Kurepa Hypothesis Prikry-type forcing AIM forcing club_AD Whitehead Problem Monotonically far Absoluteness Reflecting stationary set Reduced Power Ulam matrix Fat stationary set diamond star Mandelbrot set polarized partition relation sap Non-saturation Cardinal function Analytic sets Knaster Sigma-Prikry nonmeager set Diamond for trees Distributive tree 54G20 Forcing with side conditions Vanishing levels Hedetniemi's conjecture Successor of Singular Cardinal Amenable C-sequence perfectly normal Cardinal Invariants Antichain Strongly compact cardinal Almost-disjoint family Weakly compact cardinal coloring number Closed coloring Jonsson cardinal Hereditarily Lindelöf space Rock n' Roll Aronszajn tree PFA(S)[S] Universal Sequences GMA Parameterized proxy principle Shelah's Strong Hypothesis Cohen real full tree O-space Almost countably chromatic Hindman's Theorem S-Space Club Guessing Nonspecial tree Interval topology on trees Fodor-type reflection free Boolean algebra Almost Souslin strongly bounded groups Successor of Regular Cardinal Filter reflection specializable Souslin tree Chromatic number transformations Ostaszewski square Souslin Tree Small forcing Luzin set Postprocessing function Rainbow sets reflection principles regressive Souslin tree incompactness Generalized descriptive set theory Singular cardinals combinatorics unbounded function Axiom R Ramsey theory over partitions Chang's conjecture Constructible Universe approachability ideal indecomposable filter Poset countably metacompact Strongly Luzin set Martin's Axiom Ascending path Subtle cardinal Intersection model Erdos-Hajnal graphs higher Baire space Well-behaved magma b-scale tensor product graph square Strong coloring very good scale Sakurai's Bell inequality Commutative projection system Microscopic Approach PFA Foundations Fast club ccc Local Club Condensation. Was Ulam right? Singular Density free Souslin tree Slim tree P-Ideal Dichotomy Subadditive weak square Minimal Walks
Tag Archives: ZFC construction
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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A Shelah group in ZFC
Joint work with Márk Poór. Abstract. In a paper from 1980, Shelah constructed an uncountable group all of whose proper subgroups are countable. Assuming the continuum hypothesis, he constructed an uncountable group $G$ that moreover admits an integer $n$ satisfying … Continue reading
Posted in Groups, Publications
Tagged 03E02, 03E75, 20A15, 20E15, 20F06, Jonsson cardinal, Open Access, Strong coloring, strongly bounded groups, Subadditive, ZFC construction
3 Comments
Partitioning a reflecting stationary set
Joint work with Maxwell Levine. Abstract. We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer … Continue reading
Strong failures of higher analogs of Hindman’s Theorem
Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading
Posted in Groups, Partition Relations, Publications
Tagged 03E02, 03E35, 03E75, 05A17, 05D10, 11P99, 20M14, Chang's conjecture, Commutative cancellative semigroups, Entangled linear order, Erdos Cardinal, Hindman's Theorem, Jonsson cardinal, Kurepa Hypothesis, Square-Brackets Partition Relations, Weakly compact cardinal, ZFC construction
1 Comment
Mathematics Colloquium, Bar-Ilan University, November 2013
I gave a colloquium talk at Bar-Ilan University on November 10, 2013. Title: Forcing as a tool to prove theorems Abstract: Paul Cohen celebrated solution to Hilbert’s first problem showed that the Continuum Hypothesis is independent of the usual axioms of … Continue reading
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
Transforming rectangles into squares, with applications to strong colorings
Abstract: It is proved that every singular cardinal $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$. As a … Continue reading
