The largest Erd\H{o}s-Ko-Rado sets in 2-(v,k,1) designs

TL;DR

This paper investigates the maximum size of Erd ext{o}s-Ko-Rado sets within 2-(v,k,1) designs, including Steiner systems and unitals, providing bounds and classifications for these intersecting block sets.

Contribution

It offers new bounds and classifications for Erd ext{o}s-Ko-Rado sets in Steiner systems and unitals, expanding understanding of their structure and maximum sizes.

Findings

Determined upper bounds for Erd ext{o}s-Ko-Rado sets in certain designs.

Classified maximal Erd ext{o}s-Ko-Rado sets in Steiner systems.

Provided bounds on the second-largest maximal Erd ext{o}s-Ko-Rado sets in unitals.

Abstract

An Erd\H{o}s-Ko-Rado set in a block design is a set of pairwise intersecting blocks. In this article we study Erd\H{o}s-Ko-Rado sets in 2-(v,k,1) designs, Steiner systems. The Steiner triple systems and other special classes are treated separately. For unitals we also determine an upper bound on the size of the second-largest maximal Erdos-Ko-Rado sets.