### Archives

### 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

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

# Tag Archives: Ostaszewski square

## 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

## The Ostaszewski square, and homogeneous Souslin trees

Abstract: Assume GCH and let $\lambda$ denote an uncountable cardinal. We prove that if $\square_\lambda$ holds, then this may be witnessed by a coherent sequence $\left\langle C_\alpha \mid \alpha<\lambda^+\right\rangle$ with the following remarkable guessing property: For every sequence $\langle A_i\mid i<\lambda\rangle$ … Continue reading

Posted in Publications, Souslin Hypothesis, Squares and Diamonds
Tagged 03E05, 03E35, Club Guessing, Fat stationary set, Ostaszewski square, Souslin Tree
5 Comments