Multicoloured Clique, parameterized by the number of colours, fpt-reduces to interval scheduling with eligible machine sets, parameterized by the number $m$ of machines.

Multicoloured Clique is W[1]-complete (Fellows, Hermelin, Rosamond and Vialette 2009), so this says that interval scheduling with eligible machine sets is W[1]-hard for $m$. It is therefore fixed-parameter tractable only if $\mathrm{W}[1] = \mathrm{FPT}$, which is not believed: no algorithm solves it in $f(m) \cdot \mathrm{poly}(n)$ time unless the hierarchy collapses. The third theorem of this submission shows that adding $p_{\max}$ to the parameter does make the problem tractable, so the two results together locate the boundary.

The reduction turns an instance with $k$ colours into a scheduling instance on $\binom{k}{2}+1$ machines — one for each pair of colours, which selects an edge between those two colour classes, and one validation machine. The parameter of the image depends on the parameter of the source alone, so the reduction is an fpt-reduction and not merely a correct one.