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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

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

# Tag Archives: 05C15

## 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

## Same Graph, Different Universe

Abstract. May the same graph admit two different chromatic numbers in two different universes? how about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Godel’s constructible … Continue reading

Posted in Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, approachability ideal, Chromatic number, Constructible Universe, Forcing, Ostaszewski square
10 Comments

## Hedetniemi’s conjecture for uncountable graphs

Abstract. It is proved that in Godel’s constructible universe, for every successor cardinal $\kappa$, there exist graphs $\mathcal G$ and $\mathcal H$ of size and chromatic number $\kappa$, for which the tensor product graph $\mathcal G\times\mathcal H$ is countably chromatic. … 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