The Ostaszewski square, and homogeneous Souslin trees

5 Responses to The Ostaszewski square, and homogeneous Souslin trees

1. Pingback: Young Researchers in Set Theory 2011 | Assaf Rinot

2. saf says:

   April 23, 2013 at 8:56 am

   Submitted to Israel Journal of Mathematics, May 2011.
   Accepted April 2013.

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3. saf says:

   March 28, 2016 at 12:48 pm

   Correction: In Theorem 1.2, where I wrote “implicit in [17]” – the correct reference is not [17], but this paper.

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4. saf says:

   June 2, 2016 at 2:48 pm

   In a recent paper, Cody and Eskew introduce the following.
   Definition. A sequence $⟨a_{\alpha }\mid \alpha <\kappa ⟩$ is called a fat diamond sequence (${⧫}_{\kappa }$-sequence) if for every $X\subseteq \kappa$, $\{\alpha <\kappa \mid X\cap \alpha =a_{\alpha }\}$ is a fat stationary set.

   Note that by Theorem D of our paper, if $\lambda$ is singular, ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }$ holds and $2^{\lambda }={\lambda }^{+}$, then there exists a ${⧫}_{{\lambda }^{+}}$-sequence.
   Of course, $2^{\lambda }={\lambda }^{+}$ is necessary for the existence of a ${⧫}_{{\lambda }^{+}}$-sequence, but if one just needs a partition of $lambd{a}^{+}$ into ${\lambda }^{+}$ many pairwise disjoint fat stationary sets, then ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }$ suffices (including the case that $\lambda$ is regular), as remarked in here.

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   • Asaf Karagila says:

     June 3, 2016 at 4:44 am

     Shame for the notation. It should have been preserved for a “black diamond”. It would have gone well with Moti Gitik’s “piste”.

     

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