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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

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

# 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