Keevash–Saks–Sudakov–Verstraëte rainbow Turán conjecture for critical graphs

Let $r\ge 3$ and let $H$ be an $r$-critical graph with $h$ edges. For a multiset of $k$ graphs on the common vertex set $[n]$, write $\operatorname{ex}_k(n,H)$ for the maximum total number of edges among rainbow $H$-free systems. Let $T_{r-1}(n)$ be the Turán graph and $t_{r-1}(n)=|T_{r-1}(n)|$. Also, $t\mathcal{K}_n$ and $t\overline{\mathcal{K}}_n$ denote multisets consisting of $t$ copies of $K_n$ and $\overline{K}_n$, respectively, while $t\mathcal{T}_{r-1}(n)$ denotes $t$ copies of $T_{r-1}(n)$.

Keevash–Saks–Sudakov–Verstraëte conjecture. Suppose $k\ge h$ and $n$ is sufficiently large. Then

$$\operatorname{ex}_k(n,H)=\max\left\{k\,t_{r-1}(n),(h-1)\binom{n}{2}\right\}.$$

Moreover, $(h-1)\mathcal{K}_n\cup(k-h+1)\overline{\mathcal{K}}_n$ and $k\mathcal{T}_{r-1}(n)$ are the only extremal structures when $n$ is sufficiently large.

The conjecture has been confirmed when $r=3$ and when $H=K_r$, but remains open for general $r$-critical graphs.