An Evans-Style Result for Block Designs

Abstract

For positive integers $n$ and $k$ with $n \geq k$, an $(n,k,1)$-design is a pair $(V, \mathcal{B})$ where $V$ is a set of $n$ points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ called blocks such that each pair of points occur together in exactly one block. If we weaken this condition to demand only that each pair of points occur together in at most one block, then the resulting object is a partial $(n,k,1)$-design. A completion of a partial $(n,k,1)$-design $(V,\mathcal{A})$ is a (complete) $(n,k,1)$-design $(V,\mathcal{B})$ such that $\mathcal{A} \subseteq \mathcal{B}$. Here, for all sufficiently large $n$, we determine exactly the minimum number of blocks in an uncompletable partial $(n,k,1)$-design. This result is reminiscent of Evans' now-proved conjecture on completions of partial latin squares. We also prove some related results concerning edge decompositions of almost complete graphs into copies of $K_k$.