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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

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

# 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