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An application of Ramsey’s theory to partitions in groups. II. (English) Zbl 0790.05091
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 [(*)Z. Arad, G. Ehrlich, O. H. Kegel and 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 \(C_ 3\times C_ 3\times C_ 3\times C_ 3\).
05D10 Ramsey theory
20F99 Special aspects of infinite or finite groups
Full Text: Numdam EuDML
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