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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

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

# 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