Ramsey–Turán tiling conjecture for cliques

Let $r$ and $k$ be integers satisfying $1\le r\le k$ and $k\ge 3$. The Ramsey–Turán tiling function ${\bf{RTT}}_r(K_k)$ is defined as the asymptotic minimum-degree threshold for forcing a $K_k$-tiling under the relevant Ramsey–Turán independence condition. Ramsey–Turán tiling conjecture for cliques.

$${\bf{RTT}}_r(K_k)=1-\frac{r}{k}.$$

This conjecture proposes the exact value of the Ramsey–Turán tiling function for every clique size $k\ge 3$ and every parameter $1\le r\le k$. The paper presents it as an open question in the concluding remarks; its general validity is not established by the results discussed there.