Grounded L-graphs are polynomially $chi$-bounded

A grounded L-graph is the intersection graph of a collection of L shapes whose topmost points belong to a common horizontal line. We prove that every grounded L-graph with clique number $omega$ has chromatic number at most $17omega^4$. This improves the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is polynomially $chi$-bounded. We also survey $chi$-boundedness problems for grounded geometric intersection graphs and give a high-level overview of recent techniques to obtain polynomial bounds.