# Graphs, groups, and more: celebrating Brian Alspach’s 80th and Dragan Marušič’s 65th birthdays

from 28 May 2018 to 1 June 2018
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# Some bounds on the number of cyclic Steiner 2-designs

## Speakers

• Dr. Emanuele BRUGNOLI

## Content

A $2-(v,k,1)$ design or, also, a Steiner $2$-design is said to be cyclic if it admits an automorphism cyclically permuting all its points. To establish the number NC$(v,k)$ of cyclic $2-(v,k,1)$ designs is in general not feasible and very little is known about this number. By playing'' with $(v,k,1)$ difference families, some lower bounds on NC$(v,k)$ are given. In particular, for primes $p=6n+1$ with $p\equiv\pm 1$ (mod $5$), a construction involving the golden ratio of $\mathbb{Z}_p$ and the Narayana cows sequence is shown to give NC$(p,3)>2^{3n/2}$.