# Rectangular square-bracket operation for successor of regular cardinals

Joint work with Stevo Todorcevic.

Extended Abstract: Consider the coloring statement $\lambda^+\nrightarrow[\lambda^+;\lambda^+]^2_{\lambda^+}$ for a given regular cardinal $\lambda$:

• In 1990, Shelah proved the above for $\lambda>2^{\aleph_0}$;
• In 1991, Shelah proved the above for $\lambda>\aleph_1$;
• In 1997, Shelah proved the above for $\lambda=\aleph_1$;
• In 2006, Moore proved the above for $\lambda=\aleph_0$.

In this paper, we provide a uniform proof of the fact that $\lambda^+\nrightarrow[\lambda^+;\lambda^+]^2_{\lambda^+}$ holds for every regular cardinal $\lambda$.

Citation information:

A. Rinot and S. Todorcevic, Rectangular square-bracket operation for successor of regular cardinals, Fund. Math., 220(2): 119-128, 2013.

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### 2 Responses to Rectangular square-bracket operation for successor of regular cardinals

1. Richard Dedekind says:

I am always impressed when someone improves Shelah’s results. Shelah is one of those people that always chase after improving their results, and relentlessly if I may add. Moore gaining on him is mighty impressive. Nice work uniformizing the proof, I will have to sit and read through this later this week!

2. saf says:

Submitted to Fundamenta Mathematicae, April 2012.
Accepted, December 2012.