Tightness of the three-color rainbow-triangle bound for unbalanced color classes

Determine whether the inequality T^2\leq 2m_rm_gm_b is tight for simple 3-edge-colored graphs when the red, green, and blue edge classes have unequal sizes.

Background

The paper recalls the sharp bound T2\leq 2m_rm_gm_b for the number T of rainbow triangles in a simple 3-edge-colored graph. Equality is attained by the properly colored K_4 construction and its balanced blowups when the three color classes are equal in size, but the authors do not resolve whether equality can occur in unbalanced regimes.

References

It remains open whether the bound is tight if the sizes of the color classes are unbalanced.

— On Colorful Kruskal--Katona Theorems  (2610.02165 - Chao et al., 1 Oct 2026) in Section 1, Introduction

One of the motivations to consider the rainbow triangle problem in is to see whether the proof for the rainbow triangle theorem can be lifted up to an improvement on the multijoints problem, although it is still currently open.

— On Colorful Kruskal--Katona Theorems  (2610.02165 - Chao et al., 1 Oct 2026) in Section 1, subsection “Connection to the joints problem”

For any $\alpha_r,\alpha_g,\alpha_b\in(0,1)$ with $\alpha_r+\alpha_g+\alpha_b=1$, determine the smallest constant $C=C(\alpha_r,\alpha_g,\alpha_b)$ so that the following is true. In a simple graph $G$ with at most $\alpha_rm$ red edges, at most $\alpha_gm$ green edges and at most $\alpha_bm$ blue edges, there are at most $(C+o(1))(\alpha_r\alpha_g\alpha_b){1/2}m{3/2}$ rainbow triangles.

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