Density threshold for disjoint copies of complete graphs in multipartite graphs

Determine how large the common lower bound α_t(c) on the bipartite densities of a t-partite graph with n vertices in each part must be to guarantee cn vertex-disjoint copies of K_t, where t is fixed and k=cn.

Background

The paper studies density conditions guaranteeing several vertex-disjoint triangles in balanced tripartite graphs and separately treats triangle-factors. It then asks for an analogue for K_t in a t-partite graph, with the number of desired vertex-disjoint copies growing linearly with the part size.

The unresolved issue is to identify the sharp function α_t(c), or equivalently the minimum common bipartite density that forces a matching of size cn among the K_t copies supported by the multipartite graph.

References

Let $t$ be fixed. Assume that $G$ is a $t$-partite graph with $n$ vertices in each class and bipartite densities at least $\alpha_t(c)$. How large does $\alpha_t(c)$ have to be in order to guarantee the existence of $k=c n$ vertex disjoint copies of $K_t$ in $G$?

— Density conditions for $k$ vertex-disjoint triangles in tripartite graphs  (2503.05218 - Guo et al., 7 Mar 2025) in Problem environment in Section 1, near the end of the Introduction