### 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

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

# 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
6 Comments

## 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
Leave a comment

## 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