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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
- Genearlizations 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

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

# 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