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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

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

# 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