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Generalized descriptive set theory middle diamond Nonspecial tree Large Cardinals Ineffable cardinal HOD Whitehead Problem Amenable C-sequence Successor of Singular Cardinal Closed coloring Subtle tree property Sakurai's Bell inequality Almost-disjoint family Ascent Path Poset Ulam matrix free Boolean algebra Diamond-sharp Selective Ultrafilter GMA Fat stationary set unbounded function Ostaszewski square Distributive tree Filter reflection Dushnik-Miller C-sequence reflection principles countably metacompact diamond star weak square Prikry-type forcing Diamond Minimal Walks Kurepa Hypothesis Sigma-Prikry SNR Rainbow sets O-space super-Souslin tree coloring number Axiom R Uniformization ZFC construction Prevalent singular cardinals Lipschitz reduction Diamond for trees projective Boolean algebra Square-Brackets Partition Relations b-scale Coherent tree stationary hitting Sierpinski's onto mapping principle Chang's conjecture Erdos Cardinal S-Space transformations very good scale Local Club Condensation. Martin's Axiom square principles Dowker space Iterated forcing Fast club Uniformly homogeneous Rado's conjecture P-Ideal Dichotomy Luzin set Uniformly coherent Cohen real Slim tree indecomposable ultrafilter Forcing Axioms Mandelbrot set Souslin Tree Hindman's Theorem Subnormal ideal Strongly Luzin set Hereditarily Lindelöf space Subtle cardinal Singular Density Jonsson cardinal PFA Club Guessing Postprocessing function Small forcing Singular cofinality Greatly Mahlo incompactness Precaliber Well-behaved magma Open Access square xbox approachability ideal Forcing Almost countably chromatic sap Parameterized proxy principle Subadditive Partition Relations Commutative cancellative semigroups PFA(S)[S] ccc higher Baire space L-space Shelah's Strong Hypothesis Foundations Microscopic Approach Chromatic number club_AD free Souslin tree nonmeager set Ramsey theory over partitions Singular cardinals combinatorics Was Ulam right full tree Erdos-Hajnal graphs polarized partition relation Universal Sequences Constructible Universe Absoluteness Fodor-type reflection Reduced Power 54G20 Non-saturation Cardinal function Vanishing levels strongly bounded groups Reflecting stationary set Generalized Clubs Knaster and friends Weakly compact cardinal tensor product graph Almost Souslin Antichain AIM forcing Cardinal Invariants regressive Souslin tree Hedetniemi's conjecture Successor of Regular Cardinal OCA weak diamond Analytic sets Aronszajn tree Rock n' Roll Knaster stick positive partition relation Strong coloring specializable Souslin tree stationary reflection
Tag Archives: b-scale
6th European Set Theory Conference, July 2017
I gave a 3-lecture tutorial at the 6th European Set Theory Conference in Budapest, July 2017. Title: Strong colorings and their applications. Abstract. Consider the following questions. Is the product of two $\kappa$-cc partial orders again $\kappa$-cc? Does there exist … Continue reading
Posted in Invited Talks, Open Problems
Tagged b-scale, Cohen real, Luzin set, Minimal Walks, Souslin Tree, Square-Brackets Partition Relations
4 Comments
Open coloring and the cardinal invariant $\mathfrak b$
Nik Weaver asked for a direct proof of the fact that Todorcevic’s axiom implies the failure of CH fails. Here goes. Notation. For a set $X$, we write $[X]^2$ for the set of unordered pairs $\{ \{x,x’\}\mid x,x’\in X, x\neq … Continue reading
The S-space problem, and the cardinal invariant $\mathfrak b$
Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic … Continue reading
The S-space problem, and the cardinal invariant $\mathfrak b$
Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic … Continue reading
c.c.c. vs. the Knaster property
After my previous post on Mekler’s characterization of c.c.c. notions of forcing, Sam, Mike and myself discussed the value of it . We noticed that a prevalent verification of the c.c.c. goes like this: given an uncountable set of conditions, … Continue reading
Dushnik-Miller for regular cardinals (part 2)
In this post, we shall provide a proof of Todorcevic’s theorem, that $\mathfrak b=\omega_1$ implies $\omega_1\not\rightarrow(\omega_1,\omega+2)^2$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading
Posted in Blog, Expository
Tagged b-scale, Dushnik-Miller, Partition Relations, Square-Brackets Partition Relations
5 Comments
Infinite Combinatorial Topology
Back in 2005, as a master student, I attended a course by Boaz Tsaban, entitled “Infinite Combinatorial Topology”. A friend and I decided to produce lecture notes, but in a somewhat loose sense, that is: we sometimes omit material given … Continue reading
Posted in Notes
Tagged b-scale, Cardinal function, Cardinal Invariants, Hereditarily Lindelöf space
8 Comments