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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

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

# Category Archives: Infinite Graphs

## 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