# Assaf Rinot: Complicated colorings for double successors

Forcing Seminar (Tel-Aviv University)

Wednesday, 01/Jan/2014, 9-11.
Room 209, Schriber building, Tel-Aviv University.

Speaker: Assaf Rinot

Title: Complicated colorings for double successors

Abstract: Let $Pr_1(\lambda,\kappa)$ assert the existence of a coloring $c:[\lambda]^2\rightarrow\lambda$ with the property that for every sequence $\langle t_\alpha \mid \alpha<\lambda\rangle$ of pairwise disjoint elements of $[\lambda]^{<\kappa}$, and every $\gamma<\lambda$, there exists $\alpha<\beta<\lambda$ with $c[t_\alpha\times t_\beta]=\{\gamma\}$.

In [Sh:572], Shelah proved that $Pr_1(\lambda^{++},\lambda)$ holds for every regular cardinal $\lambda$. We shall give a proof of the latter, and will extend it to show that $Pr_1(\lambda^{++},\text{cf}(\lambda))$ holds for every singular cardinal $\lambda$ that satisfies a certain (very) weak hypothesis.