Twins in graphs

A basic pigeonhole principle insures an existence of two objects of the same type if the number of objects is larger than the number of types. Can such a principle be extended to a more complex combinatorial structure? Here, we address such a question for graphs. We call two disjoint subsets A,BA,B of vertices twins   if they have the same cardinality and induce subgraphs of the same size. Let t(G)t(G) be the largest kk such that GG has twins on kk vertices each. We provide the bounds on t(G)t(G) in terms of the number of edges and vertices using discrepancy results for induced subgraphs. In addition, we give conditions under which t(G)=|V(G)|/2t(G)=|V(G)|/2 and show that if GG is a forest then t(G)≥|V(G)|/2−1t(G)≥|V(G)|/2−1.