On Colorful Kruskal--Katona Theorems

What is the maximum number of rainbow triangles in an edge-colored graph with $m$ edges and $r$ colors? Using entropic techniques, we prove an upper bound of $C_r{m}^{3\mathrm{/}2}$ rainbow triangles with $C_r=\sqrt{\frac{2(r-2)}{9r}}$; this constant is best possible whenever there exists an affine plane of order $r-1$. We also show that constructions attaining at least $(C_r-{\epsilon }_r)m^{3\mathrm{/}2}$ rainbow triangles must exhibit an affine plane structure, which further improves the upper bound if no such affine plane exists. We also consider the problem of counting properly edge-colored cliques of larger sizes. Surprisingly, if the number $r$ of colors is odd, this count is instead maximized by blowups of a properly edge-colored $K_{r+1}$.