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A theorem of de Bruijn and Erd\\H{o}s gives \\[\n  \\limsup_{n\\to\\infty}\\frac{M_n^{(r)}}{m_n^{(r)}}\\geq 1+\\frac1r . \\] The case $r=1$ is sharp and gives the classical factor $2$. The cases $r\\geq 2$ remain much less understood. 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