Malykhin and Shapiro proved under GCH that if alpha^omega>alpha, then every infinite, totally bounded topological group of weight alpha has a non-trivial convergent sequence. A look at their proof shows in ZFC that if alpha is a strong limit cardinal of countable cofinality, then every totally bounded topological group of weight alpha has the property in the title. Using this version of their theorem, we prove that for such alpha , every compact group of weight alpha has a proper, dense, countably compact subgroup with the property in the title. We also prove that for such alpha there exists a pseudocompact, non-countably compact group of weight alpha with the property in the title and a totally bounded non-pseudocompact example for every such alpha . We prove that under certain assumptions, the existence of a totally bounded group of weight alpha with no convergent sequences is equivalent to the condition alpha^omega=alpha.

We introduce the following concept. Let G be an infinite compact group and let p be a partition of G. Then a convergent sequence (x_n) in G is said to be p-trivial if there exists a k<omega and an A in p such that for all n>=k, we have x_n in A.

Theorem: Let G be a compact Abelian group of weight >=c with Pontryagin dual Ghat such that r_0(Ghat)<|Ghat|. Then there exists a proper, dense, pseudocompact subgroup H of G and a partition of G into Haar measurable sets of measure zero such that every convergent sequence in H with a limit in H is p-trivial.

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