  The standard model and the bandit model are two generalizations of the
mistake-bound model to online multiclass classification. In both models the
learner guesses a classification in each round, but in the standard model the
learner recieves the correct classification after each guess, while in the
bandit model the learner is only told whether or not their guess is correct in
each round. For any set $F$ of multiclass classifiers, define $opt_{std}(F)$
and $opt_{bandit}(F)$ to be the optimal worst-case number of prediction
mistakes in the standard and bandit models respectively.
  Long (Theoretical Computer Science, 2020) claimed that for all $M > 2$ and
infinitely many $k$, there exists a set $F$ of functions from a set $X$ to a
set $Y$ of size $k$ such that $opt_{std}(F) = M$ and $opt_{bandit}(F) \ge (1 -
o(1))(|Y|\ln{|Y|})opt_{std}(F)$. The proof of this result depended on the
following lemma, which is false e.g. for all prime $p \ge 5$, $s = \mathbf{1}$
(the all $1$ vector), $t = \mathbf{2}$ (the all $2$ vector), and all $z$.
  Lemma: Fix $n \ge 2$ and prime $p$, and let $u$ be chosen uniformly at random
from $\left\{0, \dots, p-1\right\}^n$. For any $s, t \in \left\{1, \dots,
p-1\right\}^n$ with $s \neq t$ and for any $z \in \left\{0, \dots,
p-1\right\}$, we have $\Pr(t \cdot u = z \mod p \text{ } | \text{ } s \cdot u =
z \mod p) = \frac{1}{p}$.
  We show that this lemma is false precisely when $s$ and $t$ are multiples of
each other mod $p$. Then using a new lemma, we fix Long's proof.
