Resolvability of infinite designs

In this paper we examine the resolvability of infinite designs. We show that in stark contrast to the finite case, resolvability for infinite designs is fairly commonplace. We prove that every t  -(v,k,Λ)$(v,k,\Lambda )$ design with t finite, v   infinite and k,λ<v$k,\lambda <v$ is resolvable and, in fact, has α   orthogonal resolutions for each α<v$\alpha <v$. We also show that, while a t  -(v,k,Λ)$(v,k,\Lambda )$ design with t and λ finite, v   infinite and k=v$k=v$ may or may not have a resolution, any resolution of such a design must have v parallel classes containing v   blocks and at most λ−1$\lambda -1$ parallel classes containing fewer than v   blocks. Further, a resolution into parallel classes of any specified sizes obeying these conditions is realisable in some design. When k<v$k<v$ and λ=v$\lambda =v$ and when k=v$k=v$ and λ is infinite, we give various examples of resolvable and non-resolvable t  -(v,k,Λ)$(v,k,\Lambda )$ designs.