### Archives

### 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

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

# 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