A proof of the Jungnickel-Tonchev conjecture on quasi-multiple quasi-symmetric designs

Abstract

A proof of the following conjecture of Jungnickel and Tonchev on quasi-multiple quasi-symmetric designs is given: Let D be a design whose parameter set (v',b',r',k',λ') equals (v,sv,sk,k, sλ) for some positive integer s and for some integers v,k,λ that satisfy λ(v-1) = k(k-1) (that is, these integers satisfy the parametric feasibility conditions for a symmetric (v,k,λ)-design). Further assume that D is a quasi-symmetric design, that is D has at most two block intersection numbers. If (k,λ (s-1)) = 1, then the only way D can be constructed is by taking multiple copies of a symmetric (v,k,λ)-design.