Polynomial-time solvability of RCSPs without an rpp-construction of 3-colourability

Determine whether, for every finite structure H and every possibly infinite structure R, failure of the restricted CSP template (H,R) to rpp-construct the 3-colouring template implies that RCSP(H,R) is polynomial-time solvable.

Background

Restricted CSPs generalize CSPs with promised input classes. The paper proves a P-versus-NP-hard dichotomy when both the template and restriction are finite, and proves related results for restrictions definable in GMSNP.

The proposed question asks whether the algebraic criterion of not rpp-constructing 3-colourability characterizes tractability for arbitrary, possibly infinite restrictions.

References

Is it true that for every finite structure $$ and every (possibly infinite) $$, if $(,)$ does not rpp-construct $(_3,)$, then $\RCSP(,)$ is polynomial-time solvable? (Compare to Theorem~\ref{thm:finite-RCSP-dichotomy}).

— Restricted CSPs and F-free Digraph Algorithmics  (2502.17596 - Guzmán-Pro et al., 24 Feb 2025) in Conclusion and outlook, Section 8