A set \(H\) is called a Hilbert cube of dimension \(k\) (\(k\)-cube) if there exist positive integers \(a_ 1,\dots,a_ k\) and a non-negative integer \(a_ 0\) such that \[ H=\{a_ 0+\varepsilon_ 1 a_ 1+\cdots+\varepsilon_ ka_ k\mid \varepsilon_ i\in\{0,1\}\}. \] The author gives a new upper bound for the largest size of subset of \(\{1,2,\cdots ,n\}\) not containing a \(k\)-cube: for every \(k\geq 3\), \[ H_ k(n)\leqslant n^ {1-1/2^ {k-1}}+2n^ {1-1/2^ {k-2}}. \] Thereby he improves results of Szemerédi, Rödl and Gunderson.