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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

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

# 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