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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

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

# 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