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It is consistent that there exists a set mappingF with ß<F(ß, a)<a for ß + 2 = a >w 2 with no uncountable free sets.

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We analyze the strong polarized partition relation with respect to several cardinal characteristics and forcing notions of the reals. We prove that random reals (as well as the existence of real-valued measurable cardinals) yield downward negative polarized relations.

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