An application of Ramsey's theory to partitions in groups. II (Q1313424)

Define a group \(G\) (or partial semigroup \(G\)) to have an \(n\)-partition; in short, \(G\) is in the class \(\mathbf{nP}\), if there exists a partition of the set \(G\) into subsets \(\{1\}\), \(A_ 1,\dots,A_ n\), \(n\geq 2\), (\(A_ i\) may be empty) such that if \(x,y\in A_ i\), \(x\neq y\), \(1\leq i\leq n\), then \(xy\notin A_ i\). We proved in Part I [(*)\textit{Z. Arad}, \textit{G. Ehrlich}, \textit{O. H. Kegel} and \textit{J. C. Lennox}, Rend. Semin. Mat. Univ. Padova 84, 143-157 (1990; Zbl 0724.05071)] that infinite groups are not in \(\mathbf{nP}\), for any positive integer \(n\geq 2\). Also finite groups of order greater than \(R(2,8,(1/2)(n^ 2+2))\) are not in \(\mathbf{nP}\). In particular, we proved that for \(n=2\) groups of order greater than 9 are not in \(\mathbf{2P}\) and that for \(n=3\) groups of order greater than 18 are not in \(\mathbf{3P}\). The Ramsey numbers \(R(2,8,(1/2)(n^ 2+2))\) are large and it is an open question how to compute them. For a more complete background and more detailed information see \((*)\). The goal of this paper is to obtain the following main theorem: An abelian group \(G\) is \(\mathbf{4P}\) if and only if \(G\) is isomorphic to one of the following: (a) A cyclic group \(C_ k\) of order \(k\) where either \(k\leq 45\) or \(k=48\). (b) A noncyclic group of order \(\leq 40\). (c) A noncylcic group of order 45, 48 or 49. (d) The noncyclic group \(C_ 3\times C_ 3\times C_ 3\times C_ 2\). (e) The noncyclic group \(C_ 3\times C_ 3\times C_ 3\times C_ 3\).