On the Turán numbers of kKr in ℓ-partite graphs

Given graphs G and H, the Turán numberex(G,H)ofHinG is the maximum number of edges in a subgraph of G that contains no H. Chen et al. determined ex(Kϱ1,ϱ2,kK2) for all 1≤k≤ϱ1≤ϱ2. De Silva et al. determined ex(Kϱ1,…,ϱr,kKr) for all r≥2 and 1≤k≤ϱ1≤⋯≤ϱr. Moreover, De Silva et al. proposed an interesting generalization of ex(Kϱ1,…,ϱr,kKr): Determine ex(Kϱ1,…,ϱℓ,kKr) for ℓ≥r. In this paper, we give a proof of ex(Kϱ1,…,ϱℓ,kK2)=(k−1)∑i=2ℓϱi for all ℓ≥2 and 1≤k≤ϱ1≤⋯≤ϱℓ. We also determine the Turán numbers ex(Kϱ1,ϱ2,ϱ3,ϱ4,kK3) for all k≥1 and ϱ4≥ϱ3≥ϱ2≥ϱ1≥4(k−1), which gives a positive solution to a problem due to De Silva et al.