Sets of type-(1,n)(1,n) in symmetric designs for λ≥3λ≥3

A set of type-(m,n)S is a set of points of a design with the property that each block of the design meets either mm points or nn points of SS. The notions of type and of parameters of a kk-set (there called characters) were introduced for the first time by Tallini Scafati in [M. Tallini Scafati, {k,n}{k,n}-archi di un piano grafico finito, con particolare riguardo a quelli con due caratteri. Note I and Note II, Rend. Accad. Naz. Lincei 40 (8) (1996) 812–818 (1020–1025)]. If m=1m=1, SS gives rise to a subdesign of the design. Under weaker conditions for the order of each symmetric design, the parameters of sets of type-(1,n)(1,n) in projective planes were characterised by G. Tallini and the biplane case was dealt with by S. Kim, by solving the corresponding Diophantine equation for each case, separately. In this paper, we first characterise the parameters of sets of type-(1,n)(1,n) in the triplane with more generalised order conditions than prime power order. Next, we generalise the result on triplanes to arbitrary symmetric designs for λ≥3λ≥3. As results, a non-existence condition for special parameter sets and a characterisation of parameters for the existence, restricted by some derived bounds, are given.