SCHWEITZER 2017 Problem 4

Let $K$ be a number field which is neither $\mathbb{Q}$ nor a quadratic imaginary extension of $\mathbb{Q}$ . Denote by $\mathcal{L}(K)$ the set of integers $n\ge 3$ for which we can find units $\varepsilon_1,\ldots,\varepsilon_n\in K$ for which $$\varepsilon_1+\dots+\varepsilon_n=0,$$ but $\sum_{i\in I}\varepsilon_i\neq 0$ for any nonempty proper subset $I$ of $\{1,2,\dots,n\}$ . Prove that $\mathcal{L}(K)$ is infinite, and that its smallest element can be bounded from above by a function of the degree and discriminant of $K$ . Further, show that for infinitely many $K$ , $\mathcal{L}(K)$ contains infinitely many even and infinitely many odd elements.