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Dietzfelbinger, Hromkovi{\\v{c}}, and Schnitger (1996) showed that $n \\le (\\mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked whether the exponent on $\\mbox{rk} M$ can be improved.\n  We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size $n = \\binom{\\mbox{rk} M+1}{2}$. 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