The triple rendezvous time of a synchronizing automaton can be floor(4n/3)

RESUMO

For every $n\ge9$, we construct a strongly connected synchronizing automaton with two input letters and $n$ states whose shortest word merging some three distinct states has length $\lfloor4n/3\rfloor$. One letter is a permutation; the other has an image of size $n-1$. We give complete transition maps and an explicit word attaining the bound. For the lower bound, we assign an integer to each unordered pair of states and prove that applying either input letter can decrease it by at most one. For $n\not\equiv0\pmod3$, the same construction contains a specified state whose shortest merging word, with its partner chosen freely, has length $\lfloor4n/3\rfloor-1$.