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