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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
- 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
- Partitioning the club guessing January 22, 2014

### Keywords

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

# 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 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