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We construct several topological groups with very strong combinatorialproperties. In particular, we give simple examples of subgroups of the realline R (thus strictly o-bounded) which have the Hurewicz property but are notsigma-compact, and show that the product of two o-bounded subgroups of R^N mayfail to be o-bounded, even when they satisfy the stronger propertyS1(Borel_Omega,Borel_Omega). This solves a problem of Tkacenko and Hernandez,and extends independent solutions of Krawczyk and Michalewski and of Banakh,Nickolas, and Sanchis. We also construct separable metrizable groups G of sizecontinuum such that every countable Borel omega-cover of G contains agamma-cover of G.Comment: Small update

𝑜-bounded groups and other topological groups with strong combinatorial properties