  The Dvoretzky--Kiefer--Wolfowitz (DKW) inequality says that if $F_n$ is an
empirical distribution function for variables i.i.d.\ with a distribution
function $F$, and $K_n$ is the Kolmogorov statistic
$\sqrt{n}\sup_x|(F_n-F)(x)|$, then there is a finite constant $C$ such that for
any $M>0$, $\Pr(K_n>M) \leq C\exp(-2M^2).$ Massart proved that one can take C=2
(DKWM inequality) which is sharp for $F$ continuous. We consider the analogous
Kolmogorov--Smirnov statistic $KS_{m,n}$ for the two-sample case and show that
for $m=n$, the DKW inequality holds with C=2 if and only if $n\geq 458$. For
$n_0\leq n<458$ it holds for some $C>2$ depending on $n_0$.
  For $m\neq n$, the DKWM inequality fails for the three pairs $(m,n)$ with
$1\leq m < n\leq 3$. We found by computer search that for $n\geq 4$, the DKWM
inequality always holds for $1\leq m< n\leq 200$, and further that it holds for
$n=2m$ with $101\leq m\leq 300$. We conjecture that the DKWM inequality holds
for pairs $m\leq n$ with the $457+3 =460$ exceptions mentioned.
