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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

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

# 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