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### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Generalizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
- Happy new jewish year! September 24, 2014
- Square principles April 19, 2014

### Keywords

PFA free Boolean algebra Hindman's Theorem Fat stationary set Rado's conjecture approachability ideal incompactness Parameterized proxy principle PFA(S)[S] Cohen real Erdos Cardinal Microscopic Approach super-Souslin tree Large Cardinals stationary reflection Successor of Regular Cardinal very good scale Singular Density tensor product graph Singular coﬁnality Aronszajn tree Almost-disjoint famiy Forcing Axioms Foundations Antichain Generalized Clubs Sakurai's Bell inequality S-Space P-Ideal Dichotomy Fodor-type reflection Forcing b-scale Jonsson cardinal Distributive tree free Souslin tree Chromatic number middle diamond Constructible Universe Universal Sequences Chang's conjecture Almost Souslin Weakly compact cardinal Luzin set Postprocessing function Shelah's Strong Hypothesis Non-saturation Nonspecial tree Slim tree diamond star Small forcing HOD Ascent Path L-space Square-Brackets Partition Relations Prikry-type forcing sap weak square Almost countably chromatic Erdos-Hajnal graphs Poset Selective Ultrafilter Uniformly coherent Cardinal Invariants polarized partition relation Axiom R Successor of Singular Cardinal Whitehead Problem Martin's Axiom Prevalent singular cardinals Hereditarily Lindelöf space Mandelbrot set Rock n' Roll Ostaszewski square 11P99 ccc Partition Relations Diamond reflection principles Club Guessing Absoluteness Stevo Todorcevic Uniformization stationary hitting 05A17 Commutative cancellative semigroups Reduced Power Rainbow sets Coherent tree Singular cardinals combinatorics OCA Hedetniemi's conjecture square principles xbox coloring number Knaster weak diamond Cardinal function Dushnik-Miller specializable Souslin tree square Souslin Tree Fast club Minimal Walks Kurepa Hypothesis projective Boolean algebra

# Category Archives: Compactness

## The eightfold way

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading

Posted in Compactness
Tagged approachability ideal, Aronszajn tree, stationary reflection, Weakly compact cardinal
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## Reflection on the coloring and chromatic numbers

Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading

## Chromatic numbers of graphs – large gaps

Abstract. We say that a graph $G$ is $(\aleph_0,\kappa)$-chromatic if $\text{Chr}(G)=\kappa$, while $\text{Chr}(G’)\le\aleph_0$ for any subgraph $G’$ of $G$ of size $<|G|$. The main result of this paper reads as follows. If $\square_\lambda+\text{CH}_\lambda$ holds for a given uncountable cardinal $\lambda$, … Continue reading

Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Almost countably chromatic, Chromatic number, incompactness, Ostaszewski square
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## A topological reflection principle equivalent to Shelah’s strong hypothesis

Abstract: We notice that Shelah’s Strong Hypothesis (SSH) is equivalent to the following reflection principle: Suppose $\mathbb X$ is an (infinite) first-countable space whose density is a regular cardinal, $\kappa$. If every separable subspace of $\mathbb X$ is of cardinality at most … Continue reading

Posted in Compactness, Publications, Topology
Tagged 03E04, 03E65, 54G15, Shelah's Strong Hypothesis
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## Openly generated Boolean algebras and the Fodor-type reflection principle

Joint work with Sakaé Fuchino. Abstract: We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is $\aleph _2$-projective. Previously it was known that this … Continue reading