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### Recent blog posts

- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Genearlizations 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
- Partitioning the club guessing January 22, 2014

### Keywords

L-space Coherent tree Stevo Todorcevic Chromatic number projective Boolean algebra Ostaszewski square Foundations polarized partition relation xbox 05A17 11P99 Large Cardinals Erdos-Hajnal graphs OCA Singular cardinals combinatorics Mandelbrot set Poset Hindman's Theorem Partition Relations Jonsson cardinal 05D10 Rado's conjecture Souslin Tree Almost Souslin Singular coﬁnality Commutative cancellative semigroups Prevalent singular cardinals Selective Ultrafilter Minimal Walks Prikry-type forcing Fat stationary set Uniformization Martin's Axiom Knaster Slim tree coloring number Non-saturation Erdos Cardinal middle diamond Fast club S-Space tensor product graph weak square incompactness HOD Singular Cofinality Dushnik-Miller P-Ideal Dichotomy square sap Sakurai's Bell inequality Cardinal function Rock n' Roll very good scale Fodor-type reflection Weakly compact cardinal Square-Brackets Partition Relations free Boolean algebra Cardinal Invariants Shelah's Strong Hypothesis PFA Microscopic Approach weak diamond Reduced Power Rainbow sets Hedetniemi's conjecture Universal Sequences Antichain diamond star Absoluteness 20M14 PFA(S)[S] Ascent Path Small forcing Almost-disjoint famiy Almost countably chromatic Axiom R Club Guessing Cohen real Forcing Hereditarily Lindelöf space Singular Density Diamond approachability ideal Whitehead Problem Forcing Axioms reflection principles Kurepa Hypothesis Chang's conjecture stationary reflection Generalized Clubs Successor of Regular Cardinal Aronszajn tree Successor of Singular Cardinal ccc Constructible Universe Parameterized proxy principle b-scale stationary hitting

# 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

## Reflection on the coloring and chromatic numbers

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