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- A strong form of König’s lemma October 21, 2017
- 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
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Non-saturation Aronszajn tree stationary reflection Weakly compact cardinal Small forcing coloring number Stevo Todorcevic Commutative cancellative semigroups Partition Relations Hereditarily Lindelöf space Coherent tree diamond star Erdos-Hajnal graphs Ascent Path OCA Diamond stationary hitting Forcing Axioms Constructible Universe reflection principles tensor product graph Antichain Prevalent singular cardinals Rado's conjecture Rock n' Roll polarized partition relation Singular Density Prikry-type forcing HOD Ostaszewski square incompactness Chromatic number Postprocessing function free Boolean algebra Singular coﬁnality Kurepa Hypothesis Large Cardinals Reduced Power weak diamond Cohen real Parameterized proxy principle Singular cardinals combinatorics Nonspecial tree b-scale square principles Luzin set Cardinal Invariants Fodor-type reflection very good scale L-space Club Guessing middle diamond Poset Hindman's Theorem Foundations Jonsson cardinal Square-Brackets Partition Relations Fast club Minimal Walks Uniformization Dushnik-Miller Sakurai's Bell inequality Souslin Tree Axiom R Erdos Cardinal Almost countably chromatic Fat stationary set Absoluteness Distributive tree Whitehead Problem Rainbow sets Cardinal function projective Boolean algebra 05A17 P-Ideal Dichotomy Universal Sequences S-Space super-Souslin tree Almost Souslin Mandelbrot set weak square Uniformly coherent Forcing approachability ideal PFA Successor of Regular Cardinal PFA(S)[S] Hedetniemi's conjecture specializable Souslin tree 11P99 Knaster Generalized Clubs Slim tree Microscopic Approach Martin's Axiom Chang's conjecture Successor of Singular Cardinal Almost-disjoint famiy sap Selective Ultrafilter Shelah's Strong Hypothesis free Souslin tree ccc xbox square

# 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