Exact edge-count extremal problem for rainbow cliques

Determine the maximum number of rainbow K_d subgraphs in an r-edge-colored simple graph with a prescribed number of edges, including whether the candidate constructions described in the paper are extremal.

Background

The paper’s main properly colored-clique theorem does not resolve the more restrictive rainbow-clique problem, in which all edges of the clique must have distinct colors. The authors explicitly identify this as a harder unresolved extremal problem and illustrate it through detailed comparisons for rainbow K_4s and K_6s.

References

However, a more natural question is to determine the maximum number of rainbow $K_d$'s in a simple $r$-edge-colored graph with a given number of edges. This question looks much harder in general.

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