Girã o–Lewis–Popielarz conjecture on rainbow saturation of complete graphs

Let $K_r$ be the complete graph on $r$ vertices, let $\operatorname{sat}(n,\mathcal{R}(K_r))$ denote the minimum number of edges in an $n$-vertex edge-colored graph containing no rainbow copy of $K_r$ such that adding any non-edge in any color creates a rainbow copy of $K_r$, and fix $r\geq 3$. Girã o–Lewis–Popielarz conjecture. There exists a constant $C_r$ depending only on $r$ such that, for every $n\geq 2(r-2)$,

$$\operatorname{sat}(n,\mathcal{R}(K_r))=2(r-2)n+C_r.$$

The conjecture predicts an exact linear formula for rainbow saturation with infinitely many available colors. It is refuted: for every $r\geq 3$, the paper establishes constants $\alpha_r$ satisfying $r+\Omega(r^{1/3})\leq\alpha_r\leq r+r^{1/2}$ and $\operatorname{sat}(n,\mathcal{R}(K_r))=\alpha_r n+O(1)$, contradicting the proposed slope $2(r-2)$.