Extremal graph for the rainbow K4 pattern

Determine the extremal graph for the rainbow pattern K_4^{(6)} with s edge-colours for every s≥6, including whether the extremal graph is T_3(n) for s≥12 and K_n for s≤11.

Background

The paper records that T_3(n) is known to be extremal for the rainbow K_4 pattern when s≥5434. The authors conjecture a much sharper threshold based on comparing colourings of K_n and T_3(n), but the cases between the known bound and the conjectured threshold remain unresolved.

References

The rainbow pattern $K{(6)}_4$ for any number $s \geq 6$ of colours. It was shown in that for $s\ge 5434$ the extremal graph is $T_{3}(n)$. We conjecture, by comparing the number of 5-colourings of $K_n$ and the number of $s$-colourings of $T_3(n)$, that this holds for $s\ge 12$, and that for $s\le 11$ the extremal graph is $K_n$.

— A framework for the generalised Erdős-Rothschild problem and a resolution of the dichromatic triangle case  (2502.12291 - Gupta et al., 17 Feb 2025) in Problem statement in Section 6.4, item 2, 'Other colour patterns'