Complicated colorings

Abstract. If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^\lambda_{\ge\kappa}$ 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 $d:[\lambda]^2\rightarrow\lambda$ 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.