### Archives

### 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

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

# 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