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 $n \rightarrow \infty$. Moreover, we prove that $$k(2r-4) \le \alpha(k,r) \le \begin{cases} (k-1)(4r-k-6) &\text{ for }r \le k \le 2r-3, \\(k-1)(2r-3) &\text{ for }k \ge 2r-3, \end{cases}$$ and show that the lower bound is tight for infinitely many values of $r$ and every $k\geq 2r-1$. 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.