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- 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

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

# 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