For \(\sigma\), a system of homogeneous linear equations over \(Q\), the rationals, \(\lambda(\sigma)\) is the minimal dimension of a vector space every \(\omega\)-decomposition of which has a solution in some part. If \(\lambda (\sigma)\) exists, it is smaller than \(\exp_ \omega (\omega)\). If \(\lambda(\sigma) \leq 2^ \omega\), then \(\lambda(\sigma)< \omega_ \omega\). There exist systems with \(\lambda(\sigma_ k)= \omega_ k\) \((k=2,3,\dots)\). \(\lambda (\sigma)\) does not exist if \(\sigma\) describes the ``infinite cube''.