### Archives

### 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

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

# 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