  We study theorems giving sufficient conditions on the vertex degrees of a
graph $G$ to guarantee $G$ is $t$-tough. We first give a best monotone theorem
when $t\ge1$, but then show that for any integer $k\ge1$, a best monotone
theorem for $t=\frac1k\le 1$ requires at least $f(k)\cdot|V(G)|$ nonredundant
conditions, where $f(k)$ grows superpolynomially as $k\rightarrow\infty$. When
$t<1$, we give an additional, simple theorem for $G$ to be $t$-tough, in terms
of its vertex degrees.
