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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 Ackermannian growth rate of the induced unordered
f-regressive Ramsey numbers. This threshold coincides with the corresponding
threshold for the standard regressive Ramsey numbers.
Our proof is based on an extension of an argument from a corresponding
proof in a paper by Kojman,Lee,Omri and Weiermann 2007.