For the set \(A= \{1,2, \ldots ,n \}\), let \(A^k= \{B \mid B \subseteq A\), \(|B| =k \}\). A coloring of \(A\) is given by a map \(g_k: A^k \to \{+1,-1 \}\). For \(B \subseteq A\), define \(g_k(B)= \sum g_k(W)\), where the sum is taken over all subsets \(W\) of \(B\) for which \(|W| =k\). Let \(H_k(n)= \min \max |g_k(B)|\), where the maximum is taken over all subsets \(B\) of \(A\) and the minimum is taken over all functions \(g_k\), defined above. The authors prove for \(k \geq 1\) and \(n\) sufficiently large that \(H_k(n)\) is bounded below by \(C_k \cdot n^{(k+1)/2}\) and bounded above by \(C'_k \cdot n^{(k+1)/2}\), where \(C_k\) and \(C'_k\) are positive absolute constants.