Induced subgraphs and tree decompositions X. Towards logarithmic treewidth for even-hole-free graphs

A generalized $t$-pyramid is a graph obtained from a certain kind of tree (a subdivided star or a subdivided cubic caterpillar) and the line graph of a subdivided cubic caterpillar by identifying simplicial vertices. We prove that for every integer $t$ there exists a constant $c(t)$ such that every $n$-vertex even-hole-free graph with no clique of size $t$ and no induced subgraph isomorphic to a generalized $t$-pyramid has treewidth at most $c(t)\log n$. This settles a special case of a conjecture of Sintiari and Trotignon; this bound is also best possible for the class. It follows that several $\mathsf{\text{NP}}$-hard problems such as $\text{textsc}StableSet$, $\text{textsc}VertexCover$, $\text{textsc}DominatingSet$ and $\text{textsc}Coloring$ admit polynomial-time algorithms on this class of graphs. Results from this paper are also used in later papers of the series, in particular to solve the full version of the Sintiari-Trotignon conjecture.