Graphs without rainbow cliques of orders four and five

Let $\mathcal{G}_n^k=\{G_1,G_2,\ldots,G_k\}$ be a multiset of graphs on vertex set $[n]$ and let $F$ be a fixed graph with edge set $F=\{e_1, e_2,\ldots, e_m\}$ and $k\ge m$. We say ${\mathcal{G}_n^k}$ is rainbow $F$-free if there is no $\{i_1, i_2,\ldots, i_{m}\}\subseteq[k]$ satisfying $e_j\in G_{i_j}$ for every $j\in[m]$. Let $\ex_k(n,F)$ be the maximum $\sum_{i=1}^{k}|G_i|$ among all the rainbow $F$-free multisets ${\mathcal{G}_n^k}$. Keevash, Saks, Sudakov, and Verstra\"ete (2004) determined the exact value of $\ex_k(n, K_r)$ when $n$ is sufficiently large and proposed the conjecture that the results remain true when $n\ge Cr^2$ for some constant $C$. Recently, Frankl (2022) confirmed the conjecture for $r=3$ and all possible values of $n$. In this paper, we determine the exact value of $\ex_k(n, K_r)$ for $n\ge r-1$ when $r=4$ and $5$, i.e. the conjecture of Keevash, Saks, Sudakov, and Verstra\"ete is true for $r\in\{4,5\}$.