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Distributive tree unbounded function Lipschitz reduction Uniformly homogeneous Vanishing levels ZFC construction AIM forcing Amenable C-sequence Weakly compact cardinal Successor of Regular Cardinal Prikry-type forcing Absoluteness xbox Luzin set transformations b-scale Sigma-Prikry Uniformization specializable Souslin tree Slim tree higher Baire space Rado's conjecture Strong coloring club_AD projective Boolean algebra Forcing Axioms stationary reflection C-sequence Open Access 54G20 Universal Sequences Knaster and friends HOD Prevalent singular cardinals Jonsson cardinal Diamond-sharp diamond star sap Commutative cancellative semigroups Iterated forcing Singular Density Fast club Almost countably chromatic polarized partition relation stationary hitting weak square coloring number Successor of Singular Cardinal Axiom R Cardinal function Erdos Cardinal Subnormal ideal positive partition relation Almost-disjoint family Aronszajn tree Foundations Hereditarily Lindelöf space Fodor-type reflection Ulam matrix Souslin Tree PFA(S)[S] Forcing Uniformly coherent Diamond Square-Brackets Partition Relations Hedetniemi's conjecture super-Souslin tree Generalized Clubs Rainbow sets Reflecting stationary set Singular cofinality Microscopic Approach Local Club Condensation. square Precaliber square principles P-Ideal Dichotomy Erdos-Hajnal graphs Large Cardinals Cardinal Invariants O-space Rock n' Roll Ramsey theory over partitions Subtle cardinal Sakurai's Bell inequality L-space Dushnik-Miller Selective Ultrafilter Was Ulam right Ostaszewski square Knaster Whitehead Problem Poset Antichain indecomposable ultrafilter Minimal Walks Singular cardinals combinatorics Nonspecial tree ccc PFA Filter reflection Subtle tree property Martin's Axiom Fat stationary set Kurepa Hypothesis tensor product graph Subadditive Small forcing Chromatic number Closed coloring Well-behaved magma Coherent tree S-Space Dowker space countably metacompact Greatly Mahlo weak Kurepa tree strongly bounded groups stick approachability ideal Sierpinski's onto mapping principle reflection principles Partition Relations Generalized descriptive set theory Club Guessing Non-saturation Hindman's Theorem Parameterized proxy principle Constructible Universe regressive Souslin tree Strongly Luzin set incompactness Postprocessing function OCA Chang's conjecture free Boolean algebra Ascent Path SNR Reduced Power middle diamond Analytic sets nonmeager set Cohen real Almost Souslin GMA Shelah's Strong Hypothesis weak diamond very good scale full tree Mandelbrot set Diamond for trees free Souslin tree Ineffable cardinal
Tag Archives: ZFC construction
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, 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, Preprints
Tagged 03E02, 03E75, 20A15, 20E15, 20F06, Jonsson cardinal, Strong coloring, strongly bounded groups, Subadditive, ZFC construction
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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, Erdos Cardinal, Hindman's Theorem, Jonsson cardinal, Kurepa Hypothesis, Square-Brackets Partition Relations, Weakly compact cardinal, ZFC construction
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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
