Complicated colorings

Abstract. If $\lambda ,\kappa$ are regular cardinals, $\lambda >{\kappa }^{+}$, and $E_{\ge \kappa }^{\lambda }$ admits a nonreflecting stationary set, then ${\text{Pr}}_1(\lambda ,\lambda ,\lambda ,\kappa )$ holds.

(Recall that  ${\text{Pr}}_1(\lambda ,\lambda ,\lambda ,\kappa )$ asserts the existence of  a coloring $d:[\lambda {]}^2\to \lambda$ such that for any family $\mathcal{A}\subseteq [\lambda {]}^{<\kappa }$ of size $\lambda$, consisting of pairwise disjoint sets, and every color $\gamma <\lambda$, there exist $a,b\in \mathcal{A}$ with $sup(a)<min(b)$ satisfying $d[a\times b]=\{\gamma \}$.)

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Citation information:

A. Rinot, Complicated Colorings, Math. Res. Lett., 21(6): 1367–1388, 2014.