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