Completely Transitive Designs

Abstract

We view a design $\mathcal{D}$ as a set of $k$-subsets of a fixed set $X$ of $v$ points. A $k$-subset of $X$ is at distance $i$ from $\mathcal{D}$ if it intersects some $k$-set in $\mathcal{D}$ in $k-i$ points, and no subset in more than $k-i$ points. Thus $\mathcal{D}$ determines a partition by distance of the $k$-subsets of $X$. We say $\mathcal{D}$ is completely transitive if the cells of this partition are the orbits of the automorphism group of $\mathcal{D}$ in its induced action on the $k$-subsets of $X$. This paper initiates a study of completely transitive designs $\mathcal{D}$. A classification is given of all examples for which the automorphism group is not primitive on $X$. In the primitive case the focus is on examples with the property that any two distinct $k$-subsets in $\mathcal{D}$ have at most $k-3$ points in common. Here a reduction is given to the case where the automorphism group is $2$-transitive on $X$. New constructions are given by classifying all examples for some families of $2$-transitive groups, leaving several unresolved cases.