Tight quasi-symmetric designs

TL;DR

This paper provides an elementary proof that quasi-symmetric designs without repeated blocks have at most 2 blocks, achieving equality only in the case of tight 4-designs, thus characterizing their maximal size.

Contribution

It offers a simple proof establishing an upper bound on the size of quasi-symmetric designs and characterizes when this bound is attained.

Findings

Maximum of 2 blocks in such designs

Equality characterizes tight 4-designs

Elementary proof approach

Abstract

I give an elementary proof that a quasi-symmetric design without repeated blocks on $v$ points has at most $v\choose2$ blocks, with equality if and only if it is a tight $4$-design.