  For their famous algorithm for the disjoint paths problem, Robertson and
Seymour proved that there is a function $f$ such that if the tree-width of a
graph $G$ with $k$ pairs of terminals is at least $f(k)$, then $G$ contains a
solution-irrelevant vertex (Graph Minors. XXII., JCTB 2012). We give a
single-exponential lower bound on $f$. This bound even holds for planar graphs.
