### 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

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

# 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