@inproceedings{10055,
  abstract     = {Repeated idempotent elements are commonly used to characterise iterable behaviours in abstract models of computation. Therefore, given a monoid M, it is natural to ask how long a sequence of elements of M needs to be to ensure the presence of consecutive idempotent factors. This question is formalised through the notion of the Ramsey function R_M associated to M, obtained by mapping every k ∈ ℕ to the minimal integer R_M(k) such that every word u ∈ M^* of length R_M(k) contains k consecutive non-empty factors that correspond to the same idempotent element of M. In this work, we study the behaviour of the Ramsey function R_M by investigating the regular 𝒟-length of M, defined as the largest size L(M) of a submonoid of M isomorphic to the set of natural numbers {1,2, …, L(M)} equipped with the max operation. We show that the regular 𝒟-length of M determines the degree of R_M, by proving that k^L(M) ≤ R_M(k) ≤ (k|M|⁴)^L(M). To allow applications of this result, we provide the value of the regular 𝒟-length of diverse monoids. In particular, we prove that the full monoid of n × n Boolean matrices, which is used to express transition monoids of non-deterministic automata, has a regular 𝒟-length of (n²+n+2)/2.},
  author       = {Jecker, Ismael R},
  booktitle    = {38th International Symposium on Theoretical Aspects of Computer Science},
  isbn         = {978-3-9597-7180-1},
  issn         = {1868-8969},
  location     = {Saarbrücken, Germany},
  publisher    = {Schloss Dagstuhl - Leibniz Zentrum für Informatik},
  title        = {{A Ramsey theorem for finite monoids}},
  doi          = {10.4230/LIPIcs.STACS.2021.44},
  volume       = {187},
  year         = {2021},
}