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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

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

# 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