About a question of Gateva-Ivanova and Cameron

Postato il 6 aprile 2019

April 10, 2019
11.00 – 13.00 – Aula M5
Marco Castelli

In 2011 Gateva-Ivanova and Cameron [2], following a suggestion due by P. Martin, posed the following question:

Question. For each positive integer $m$ denote by $N_m$ the minimal integer so that there exists a square-free involutive multipermutational solution $(X_m,r_m)$ of order $|X_m| = N_m$ and with $mpl(X_m, r_m) = m$ . How does $N_m$ depend on $m$ ?

They showed [2] that $N_m \geq 2^{m-1} + 1$ for every $m \in N$ and, some years later, Lebed and Vendramin improve [3] the estimation showing indirectly that $N_m \leq 2^{m-2}$ for every $m > 4$ . In this talk we will present some recent results obtained in [1] that allow us to improve the estimation of the sequence $N_m$ .