Let \({\mathbf S}\) denote the usual class of normalized, univalent functions \(f(z)=z+a_2z^2+\dots\) in the unit disk. It is known that for \(f\in {\mathbf S}\), the inequality \(0\leq\alpha (f):= \lim_{n\to\infty} |a_n|/n \leq 1\) holds. Set \(\sigma_n(f): =(a_2,\dots,a_n)\). The \(n\)th coefficient region \(V_n\subset C^{n-1}\) is the set \(V_n:=\{\sigma_n(f): f\in{\mathbf S}\}\). Suppose that \(a\in\text{int} V_N\), that the sequence \(\{\varepsilon_n\}\) of positive numbers \(\varepsilon_n\) has limit zero, and that an infinite set \(E\) of disjoint intervals of integers is given. The authors prove that there exists a function \(f\in S\) with the following properties: (i) \(\sigma_N(f)=a\); (ii) \(a_n=0(n\in I)\) for infinitely many intervals \(I\) in \(E\); and (iii) \(|a_n|>n \varepsilon_n (n\in I)\) for infinitely many intervals \(I\) in \(E\). If the coefficients \(a_n\) are real, then the inequality in (iii) can be strengthened to \(a_n>n\varepsilon_n\) \((n\in I)\). The authors do not know whether there is a function \(f\in{\mathbf S}\) with properties (i), (ii), and (iii) all of whose coefficients are nonnegative. For the prescribed initial \(N\) coefficients, the sequence \(\{\varepsilon_n\}\), and the set \(E\) of intervals of integers, the authors inductively construct the function \(f\in S\) from polynomials.