Optimal rainbow-triangle bound when no projective plane exists

Determine the optimal upper bound for the number of rainbow triangles in an r-edge-colored simple graph with m edges when r=7 or r=11, corresponding respectively to the unresolved existence of finite projective planes of orders 6 and 10.

Background

The paper proves that if no finite projective plane of order r-1 exists, then the general rainbow-triangle bound admits a saving depending on r. Since finite projective planes of orders 6 and 10 are known not to exist, the cases r=7 and r=11 are natural instances in which the exact optimal constant is not determined by the general theorem.

References

It might thus be interesting to determine the optimal bound in \cref{thm:r-color} when $r=7$ or $11$.

— On Colorful Kruskal--Katona Theorems  (2610.02165 - Chao et al., 1 Oct 2026) in Section 1, subsection “Optimal configurations and stability”

Asking the same question for more colors leads to the following natural conjecture.

— On Colorful Kruskal--Katona Theorems  (2610.02165 - Chao et al., 1 Oct 2026) in Section 7, subsection “Many colors with a given number of vertices”