Partitions of large Rado graphs


Let κ be a cardinal which is measurable after generically adding iκ+ω many Cohen subsets to κ and let G = (κ,E) be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value r m such that the set [κ] can be partitioned into classes ̇ Ci : i < r + m ̧ such that for any coloring of any of the classes Ci in fewer than κ colors, there is a copy G∗ of G in G such that [G∗]m ∩ Ci is monochromatic. It follows that G → (G)m<κ/r+ m , that is, for any coloring of [G] with fewer than κ colors there is a copy G′ of G such that [G′]m has at most r m colors. On the other hand, we show that there are colorings of G such that if G′ is any copy of G then Ci ∩ [G′]m 6= ∅ for all i < r m, and hence G 9 [G]mr+ m . We characterize r m as the cardinality of a certain finite set of types and obtain an upper and a lower bound on its value. In particular, r 2 = 2 and for m > 2 we have r m > rm where rm is the corresponding number of types for the countable Rado graph.

DOI: 10.1007/s00153-009-0138-2

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@article{Dzamonja2009PartitionsOL, title={Partitions of large Rado graphs}, author={Mirna Dzamonja and Jean A. Larson and William J. Mitchell}, journal={Arch. Math. Log.}, year={2009}, volume={48}, pages={579-606} }