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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

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