Some remarks on uncountable rainbow Ramsey theory

We discuss the rainbow Ramsey theorems at limit cardinals and successors of singular cardinals, addressing some questions by Uri Abraham, James Cummings, Clifford Smyth [J. Symbolic Logic 72 (2007), pp. 865–895] Abraham Cummings [Cent. Eur. J. Math. 10 (2012), 1004–1016]. In particular, we show for inaccessible ? \kappa , alttext="kappa right-arrow Superscript p o l y Baseline left-parenthesis kappa right-parenthesis Subscript 2 minus b d 2"> ? p o l y stretchy="false">( stretchy="false">) 2 ?<!-- ? <mml:mi>b d encoding="application/x-tex">\kappa \to ^{poly}(\kappa )^2_{2-bdd} does not characterize weak compactness any cardinal alttext="white medium square Baseline"> ? encoding="application/x-tex">\square _\kappa implies plus with stroke eta greater-than + ? ?<!-- ? <mml:mo>&gt; ^+\not ^{poly} (\eta )^2_{&gt;\kappa -bdd} alttext="eta greater-than-or-equal-to c f plus"> ?<!-- ? <mml:mi>c f encoding="application/x-tex">\eta \geq cf(\kappa )^+ equals kappa"> = ^{&gt;cf(\kappa )}=\kappa nu ?<!-- ? ^+\to (\nu alttext="nu encoding="application/x-tex">\nu &gt;cf(\kappa . also provide a simplified construction model alttext="omega omega 1 ?<!-- ? <mml:mn>1 encoding="application/x-tex">\omega _2\not (\omega _1)^2_{2-bdd} originally constructed 1004–1016] witnessing coloring is indestructible under strongly proper forcings but destructible c.c.c forcing. Finally, conclude remarks on possible generalizations to partition relations triples.