The largest Erdős–Ko–Rado sets in $$2-(v,k,1)$$ 2 - ( v , k , 1 ) designs

Abstract

An Erdős-Ko-Rado set in a block design is a set of pairwise intersecting blocks. In this article we study Erdős-Ko-Rado sets in 2 − (v, k, 1) designs, Steiner systems. The Steiner triple systems and other special classes are treated separately. For k ≥ 4, we prove that the largest Erdős-Ko-Rado sets cannot be larger than a point-pencil if r ≥ k 2 − 3k + 3 4 √ k + 2 and that the largest Erdős-Ko-Rado sets are point-pencils if also r = k 2 − k + 1 and (r, k) = (8, 4). For unitals we also determine an upper bound on the size of the second-largest maximal Erdős-Ko-Rado sets.