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We consider the problem of allocating indivisible goods to agents withadditive valuation functions. Kurokawa, Procaccia and Wang {[JACM, 2018]}present instances for which every allocation gives some agent less than hermaximin share. We present such examples with larger gaps. For three agents andnine items, we design an instance in which at least one agent does not get morethan a $\frac{39}{40}$ fraction of her maximin share. {Moreover, we show thatthere is no negative example in which the difference between the number ofitems and the number of agents is smaller than six, and that the gap (of$\frac{1}{40}$) of our example is worst possible among all instances with nineitems.}  For $n \ge 4$ agents, we show examples in which at least one agent does notget more than a $1 - \frac{1}{n^4}$ fraction of her maximin share. {In theinstances designed by Kurokawa, Procaccia and Wang, the gap is exponentiallysmall in $n$.}  Our proof techniques extend to allocation of chores (items of negativevalue), though the quantitative bounds for chores are different from those forgoods. For three agents and nine chores, we design an instance in which the MMSgap is $\frac{1}{43}$.

A tight negative example for MMS fair allocations