A Ramsey-type Theorem for Multiple Disjoint Copies of Induced Subgraphs (Extended Abstract)

Let k and ℓ be positive integers with ℓ⩽k−2. It is proved that there exists a positive integer c depending on k and ℓ such that every graph of order (2k−1−ℓ/k)n+c contains n vertex disjoint induced subgraphs, where these subgraphs are isomorphic to each other and they are isomorphic to one of four graphs: (1) a clique of order k, (2) an independent set of order k, (3) the join of a clique of order ℓ and an independent set of order k−ℓ, or (4) the union of an independent set of order ℓ and a clique of order k−ℓ.