Novák's conjecture on cyclic Steiner triple systems and its generalization

Abstract

Novák conjectured in 1974 that for any cyclic Steiner triple systems of order v with v≡1(mod6), it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic (v,k,λ)-designs with 1⩽λ⩽k−1. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all v, but we conjecture that it holds whenever v is sufficiently large compared to k. We confirm that the generalization of the conjecture holds when v is prime and λ=1 and also when λ⩽(k−1)/2 and v is sufficiently large compared to k. As a corollary, we show that for any k⩾3, with the possible exception of finitely many composite orders v, every cyclic (v,k,1)-design without short orbits is generated by a (v,k,1)-disjoint difference family.