A Ramsey-type Theorem for Multiple Disjoint Copies of Induced Subgraphs

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.