Combinatorial properties of bounded merge-width classes

Prove that every graph class of bounded merge-width is polynomially chi-bounded, has the strong Erdős–Hajnal property, and admits adjacency labelling schemes of size \(O(\log n)\).

Background

Bounded twin-width classes and structurally bounded-expansion classes share several combinatorial properties: polynomial chi-boundedness, the strong Erdős–Hajnal property, and logarithmic-size adjacency labelling schemes. Since both are examples of bounded merge-width, the paper conjectures that these properties extend to all bounded merge-width classes.