  Jamison and Sprague defined a graph $G$ to be a $k$-threshold graph with
thresholds $\theta_1 , \ldots, \theta_k$ (strictly increasing) if one can
assign real numbers $(r_v)_{v \in V(G)}$, called ranks, such that for every
pair of vertices $v,w$, we have $vw \in E(G)$ if and only if the inequality
$\theta_i \leq r_v + r_w$ holds for an odd number of indices $i$. When $k=1$ or
$k=2$, the precise choice of thresholds $\theta_1, \ldots, \theta_k$ does not
matter, as a suitable transformation of the ranks transforms a representation
with one choice of thresholds into a representation with any other choice of
thresholds. Jamison asked whether this remained true for $k \geq 3$ or whether
different thresholds define different classes of graphs for such $k$, offering
\$50 for a solution of the problem. Letting $C_t$ for $t > 1$ denote the class
of $3$-threshold graphs with thresholds $-1, 1, t$, we prove that there are
infinitely many distinct classes $C_t$, answering Jamison's question. We also
consider some other problems on multithreshold graphs, some of which remain
open.
