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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

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

# Category Archives: Infinite Graphs

## 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