Partitioning α-large sets:: Some lower bounds

Let a (beta)(m)(r) denote the property: if A is an alpha-large set of natural numbers and [A](r) is partitioned into m parts, then there exists a beta-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 alpha in terms of beta, m, r for some specific beta.