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Let α → (β) r m denote the property: if A is an α-large set of natural numbers and [A] r is partitioned into m parts, then there exists a β-large subset of A which is homogeneous for this partition. Here the notion of largeness is in the sense of the so-called Hardy hierarchy. We give a lower bound for a in terms of β, m, r for some specific β.

Partitioning 𝛼–large sets: Some lower bounds