# A notion of twins

## TL;DR

This paper introduces a generalized twin problem in combinatorial structures, improving bounds on twin sizes in graphs, strings, and permutations, and disproving a previous conjecture.

## Contribution

It proposes a new variant of the twin problem that unifies different cases and provides improved bounds and counterexamples to existing conjectures.

## Key findings

- Improved bounds on twin sizes in graphs, strings, and permutations.
- Disproved a conjecture by Dudek, Grytczuk, and Ruciński.
- Unified framework for twin problems in ordered combinatorial objects.

## Abstract

Given a combinatorial structure, a ``twin'' is a pair of disjoint substructures which are isomorphic (or look the same in some sense). In recent years, there have been many problems about finding large twins in various combinatorial structures. For example, given a graph $G$, one can ask what is the largest $s$ such that there exist disjoint subsets $I,J\subset V(G)$ on $s$ vertices, such that the induced subgraphs $G[I],G[J]$ are isomorphic. We are motivated by two different problems of finding twins in two kinds of ordered objects (strings and permutations). We introduce a new variant of ``twin problem'' which generalizes both of these. By considering this generalization, we are able to improve some bounds obtained by Dudek, Grytczuk, and Ruci\'nski, and give a negative answer to a conjecture of theirs.