  We prove that for every integer $t\geq 1$ there exists an integer $c_t\geq 1$
such that every $n$-vertex even-hole-free graph with no clique of size $t$ has
treewidth at most $c_t\log{n}$. This resolves a conjecture of Sintiari and
Trotignon, who also proved that the logarithmic bound is asymptotically best
possible. It follows that several \textsf{NP}-hard problems such as
\textsc{Stable Set}, \textsc{Vertex Cover}, \textsc{Dominating Set} and
\textsc{Coloring} admit polynomial-time algorithms on this class of graphs. As
a consequence, for every positive integer $r$, $r$-{\sc Coloring} can be solved
in polynomial time on even-hole-free graphs without any assumptions on clique
size.
  As part of the proof, we show that there is an integer $d$ such that every
even-hole-free graph has a balanced separator which is contained in the
(closed) neighborhood of at most $d$ vertices. This is of independent interest;
for instance, it implies the existence of efficient approximation algorithms
for certain \textsf{NP}-hard problems while restricted to the class of all
even-hole-free graphs.
