Erdős-Ko-Rado properties of Steiner 2-designs

Abstract

In this paper, we prove an Erdős-Ko-Rado characterisation of maximum intersecting families of blocks in Steiner $2$-designs arising from Desarguesian maximal arcs. This answers a recent question of Goryainov and Konstantinova, and implies that, among the known Steiner $2$-designs, only finitely many admit a maximum intersecting family that is neither canonical nor associated with a subdesign. We also perform a computational study of $2$-$(120,8,1)$ designs and find strong counterexamples to a problem of Godsil and Meagher. Finally, we give a parametric generalisation of $2$-$(66,6,1)$ designs with tight dual arcs as non-canonical maximum intersecting families.