For every positive integer k, we construct an explicit family of functions f: { 0, 1}ⁿ → { 0, 1} which has (k+1)-party communication complexity O(k) under every partition of the input bits into k+1 parts of equal size, and k-party communication complexity Ω(n / k⁴ 2^k) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain K_s,t as a subgraph.