On a local similarity of graphs

We say that two graphs G and H , having the same number of vertices n , are k -similar if they contain a common induced subgraph of order k . We will consider the following question: how large does n need to be to ensure at least one k -similar pair in any family of l graphs on n vertices? We will present various lower and upper bounds on n . In particular, we will prove that for l = 3 , n equals the Ramsey number R ( k , k ) . Last but not least we will determine the exact values of n for k = 3 , k = 4 and all l .