### Archives

### 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

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

# 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