Optimal constant for properly colored K4s with four colors

Determine the smallest constant C such that every 4-edge-colored simple graph with m edges contains at most Cm^2 properly colored K_4s.

Background

The paper proves a sharp formula for properly colored K_d counts in the odd-color regime and notes that the even-color case is not settled. In particular, the authors state that the correct constant for four colors is unclear and isolate the 4-colored properly colored K_4 problem as a concrete question.

References

It is likely that we could carry out a stability argument akin to the one in \cref{sec:sturctural_stability} to improve the constant by a little bit when $r$ is even, but it is unclear to us what the right constant should be in that case. To be more specific, we pose the following question. Determine the smallest constant $C$ so that the following holds. Every $4$-edge-colored simple graph with $m$ edges contains at most $Cm2$ properly colored $K_4$'s.

— On Colorful Kruskal--Katona Theorems  (2610.02165 - Chao et al., 1 Oct 2026) in Section 7, subsection “Stability for Theorem 1.8”