  A hole in a graph $G$ is an induced cycle of length at least four, and an
even hole is a hole of even length. The diamond is the graph obtained from the
complete graph $K_4$ by removing an edge. A pyramid is a graph consisting of a
triangle called the base, a vertex called the apex, and three internally
disjoint paths starting at the apex and disjoint otherwise, each joining the
apex to a vertex of the base. For a family $\mathcal{H}$ of graphs, we say a
graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to
a member of $\mathcal{H}$. Cameron, da Silva, Huang, and Vu\v{s}kovi\'c proved
that (even hole, triangle)-free graphs have treewidth at most five, which
motivates studying the treewidth of even-hole-free graphs of larger clique
number. Sintiari and Trotignon provided a construction of (even hole, pyramid,
$K_4$)-free graphs of arbitrarily large treewidth.
  Here, we show that for every $t$, (even hole, pyramid, diamond, $K_t$)-free
graphs have bounded treewidth. The graphs constructed by Sintiari and Trotignon
contain diamonds, so our result is sharp in the sense that it is false if we do
not exclude diamonds. Our main result is in fact more general, that treewidth
is bounded in graphs excluding certain wheels and three-path-configurations,
diamonds, and a fixed complete graph. The proof uses "non-crossing
decompositions" methods similar to those in previous papers in this series. In
previous papers, however, bounded degree was a necessary condition to prove
bounded treewidth. The result of this paper is the first to use the method of
"non-crossing decompositions" to prove bounded treewidth in a graph class of
unbounded maximum degree.
