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Previously only two examples of Banach space quotient maps which do not admituniformly continuous right inverses were known: one due to Aharoni andLindenstrauss and one due to Kalton ($\ell^\infty\to\ell^\infty/c_{0}$). Weshow through an application of Kalton's Monotone Transfinite Sequence Theoremthat a quotient map of a subspace of $\ell^\infty$ of sequences that convergeto zero along an ideal in $\mathbb{N}$ toward another such subspace, providedone of the ideals is `much larger' than the other, cannot have a uniformlycontinuous right inverse. We show in general that pairs of ideals in$\mathbb{N}$, with one much larger than the other, occur in abundance. Someclassical examples of ideals in $\mathbb{N}$ presented explicitly are: thefinite subsets of $\mathbb{N}$, the subsets of $\mathbb{N}$ with convergentreciprocal series, and, the subsets of $\mathbb{N}$ with density zero, Banachdensity zero or Buck density zero.

A family of quotient maps of $\ell^\infty$ that do not admit uniformly continuous right inverses