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### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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