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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 $$C_ 3\times C_ 3\times C_ 3\times C_ 3$$.
MSC:
 05D10 Ramsey theory 20F99 Special aspects of infinite or finite groups
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References:
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