Finite forbidden edge-colouring characterization of bipartite graphs

Prove or disprove that no finite set of 2-edge-coloured graphs expresses the class of bipartite graphs.

Background

The authors identify bipartite graphs as a likely example of a hereditary class that cannot be expressed using finitely many forbidden 2-edge-coloured patterns. This assertion is presented as a belief rather than a proved theorem, making it an explicit unresolved conjectural direction.

References

In particular, we believe that there is no finite set of 2-edge-coloured graphs expressing bipartite graphs, and no finite set of oriented graphs characterizing co-bipartite graphs.

— On the expressive power of $2$-edge-colourings of graphs  (2503.07409 - Bok et al., 10 Mar 2025) in Section 10, Conclusions