Ordered Gyárfás–Sumner conjecture

Prove or refute that for every fixed ordered forest F and every clique-number bound $\omega$, there exists a constant exponent $\varepsilon>0$ such that every sufficiently large n-vertex ordered graph with clique number at most $\omega$ and no induced copy of F has chromatic number at most $n^{\varepsilon}$.

Background

The paper formulates a proposed relaxation of the ordered analogue of the Gyárfás–Sumner conjecture. The unrestricted ordered analogue is false, as shown by ordered shift graphs, so the conjecture concerns subpolynomial or polynomially controlled chromatic number rather than chi-boundedness. The authors explicitly state that they do not yet have a solution; their final construction instead gives triangle-free ordered graphs avoiding a simple ordered path while having almost polynomial chromatic number.

References

While we do not have a solution to this conjecture yet, our final construction shows that there exist triangle-free families of ordered graphs that avoid a simple three-edge path as an induced subgraph, yet have almost polynomial chromatic number.

— Geometric triangle-free graphs of large chromatic number  (2610.03517 - Tomon, 2 Oct 2026) in Section 1, subsection “Forbidding ordered trees”