Abstract
Given 𝓁<k<n, a partial Steiner (n, k, 𝓁)‐system 𝒫 is a family of k‐element subsets of an n‐element set such that every 𝓁‐element subset is contained in at most one member of 𝒫. The second author, followed by several others, verified a conjecture of P. Erdős and H. Hanani stating that, for every n, there exists a partial Steiner (n, k, 𝓁)‐system of size \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$${{\left( {1 - o\left( 1 \right)} \right)\left( {_{\ell }^{n} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - o\left( 1 \right)} \right)\left( {_{\ell }^{n} } \right)} {\left( {_{\ell }^{k} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {_{\ell }^{k} } \right)}}$$\end{document}. While all known proofs involve the “semi‐random method”, in this short manuscript we use an algebraic construction (European J Combin 16(1) (1995), 35–40; Problems Control Inform Theory/Problemy Upravlen Teor Inform 12(1) (1983), 3–10) and complement it by an elementary counting argument to obtain a simple proof of the above fact. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 448–455, 2009