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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

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

# Tag Archives: 05C63

## 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