Partite Saturation of Complete Graphs

We study the problem of determining $sat(n,k,r)$, the minimum number of edges in a $k$-partite graph $G$ with $n$ vertices in each part such that $G$ is $K_r$-free but the addition of an edge joining any two non-adjacent vertices from different parts creates a $K_r$. Improving recent results of Ferrara, Jacobson, Pfender and Wenger, and generalizing a recent result of Roberts, we define a function $\alpha (k,r)$ such that $sat(n,k,r)=\alpha (k,r)n+o(n)$ as $n\to \mathrm{\infty }$. Moreover, we prove that $$k(2r-4)\le \alpha (k,r)\le \{\begin{array}{ll}(k-1)(4r-k-6)& \text{ for }r\le k\le 2r-3,\\ (k-1)(2r-3)& \text{ for }k\ge 2r-3,\end{array}$$ and show that the lower bound is tight for infinitely many values of $r$ and every $k\ge 2r-1$. This allows us to prove that, for these values, $sat(n,k,r)=k(2r-4)n+O(1)$ as $n \rightarrow \infty$. Along the way, we disprove a conjecture and answer a question of the first set of authors mentioned above.