We present a construction called layered wheel. Layered wheels are graphs of arbitrarily large treewidth and girth. They might be an outcome for a possible theorem characterizing graphs with large treewidth in terms of their induced subgraphs (while such a characterization is well-understood in terms of minors). They also provide examples of graphs of large treewidth and large rankwidth in well-studied classes, such as (theta, triangle)-free graphs and even-hole-free graphs with no K4 (where a hole is a chordless cycle of length at least four, a theta is a graph made of three internally vertex disjoint paths of length at least two linking two vertices, and K4 is the complete graph on four vertices).
@article{arxiv.1906.10998,
title = {(Theta, triangle)-free and (even hole, $K_4$)-free graphs. Part 1 : Layered wheels},
author = {Ni Luh Dewi Sintiari and Nicolas Trotignon},
journal= {arXiv preprint arXiv:1906.10998},
year = {2023}
}