[For the entire collection see Zbl 0676.00009.]
For a totally disconnected compact set E in the extended z-plane, we denote by \(M_ E\) the totality of meromorphic functions each of which is defined in the domain complementary to E and has E as the set of transcendental singularities. A function f(z) of \(M_ E\) is said to be exceptionally ramified at a singularity \(\zeta\in E\), if there exist values \(w_ k\), \(1\leq k\leq q\), and positive integers \(\nu_ k\), \(1\leq k\leq q\), with \[ \sum^{q}_{k=1}(1-(\nu_ k)^{-1})>2, \] such that, in some neighborhood of \(\zeta\), the multiplicity of any \(w_ k\)-point of f(z) is not less than \(\nu_ k\), where we set \(\nu_ k=\infty\) when f(z) has no \(w_ k\)-points there. If f(z) has three Picard exceptional values at \(\zeta\), then it is exceptionally ramified there. Our problem is to show the existence of perfect sets E with the property that each function of \(M_ E\) cannot be exceptionally ramified at any singularity \(\zeta\in E\). T. Kurokawa showed in Nagoya Math. J. 88, 133-154 (1982; Zbl 0471.30019), that Cantor sets E with successive ratios \(\{\xi_ n\}\) satisfying \(\xi_{n+1}=o(\xi^ 5_ n)\) have this property. In this paper, we show that the above condition can be replaced by a much weaker condition \(\xi_{n+1}=o(\xi^ 2).\)