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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

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

# 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