Repetitions in Graphs and Sequences

The existence of unavoidable repeated substructures is a known phenomenon implied by the pigeonhole principle and its generalizations. A fundamental problem is to determine the largest size of a repeated substructure in any combinatorial structure from a given class. The strongest notion of repetition is a pair of isomorphic substructures, such as a pair of vertexdisjoint or edge-disjoint isomorphic subgraphs or a pair of disjoint identical subsequences of a sequence. A weaker notion of repetition is a pair of substructures that have the same value on a certain set of parameters. This includes vertex-disjoint induced subgraphs of the same order and size, disjoint vertex sets with the same multiset of pairwise distances, subgraphs with the same maximum degree. This paper surveys results on unavoidable repetitions, also referred to as twins, with a focus on three asymptotically tight results obtained over the past 5 years.