Growth of χ(K_4^{Ω_w(K_m)})

Determine whether χ(K_4^G) can be much larger than 5 when G = Ω_w(K_m) and w is a sufficiently large odd integer.

Background

The paper explains that stronger lower bounds for exponential graphs could improve the upper bound on the Poljak–Rödl function. Theorem 2.?? establishes χ(K_4G) ≥ 5 for sufficiently large odd w when G = Ω_w(K_m), but the possible magnitude of this chromatic number is unknown.

References

It is unknown if $\chi(K_4G)$ can be much larger than $5$ (if $w$ is large enough).

— A survey on Hedetniemi's conjecture  (2502.16078 - Zhu, 22 Feb 2025) in Section 5, paragraph following Question q2