  A theta is a graph consisting of two non-adjacent vertices and three
internally disjoint paths between them, each of length at least two. 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 $\mathcal{H}$. We prove a
conjecture of Sintiari and Trotignon, that there exists an absolute constant
$c$ for which every (theta, triangle)-free graph $G$ has treewidth at most
$c\log (|V(G)|)$. A construction by Sintiari and Trotignon shows that this
bound is asymptotically best possible, and (theta, triangle)-free graphs
comprise the first known hereditary class of graphs with arbitrarily large yet
logarithmic treewidth.
  Our main result is in fact a generalization of the above conjecture, that
treewidth is at most logarithmic in $|V(G)|$ for every graph $G$ excluding the
so-called three-path-configurations as well as a fixed complete graph. It
follows that several NP-hard problems such as Stable Set, Vertex Cover,
Dominating Set and Coloring admit polynomial time algorithms in graphs
excluding the three-path-configurations and a fixed complete graph.
