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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

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