Following ideas of Richer (2000) we introduce the notion of unordered regressive Ramsey numbers or unordered Kanamori-McAloon numbers. We show that these are of Ackermannian growth rate. For a given number-theoretic function f we consider unordered f-regressive Ramsey numbers and classify exactly the threshold for f which gives rise to the Acker-mannian growth rate of the induced f-regressive Ramsey numbers. This threshold coincides with the corresponding threshold for the standard re-gressive Ramsey numbers. Our proof is based on an extension of an argumtent from a corresponding proof in a paper by Kojman,Lee,Omri and Weiermann 2007.
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