An Evans-style result for block designs

TL;DR

This paper determines the minimum size of uncompletable partial block designs for large n, extending Evans' conjecture on Latin squares to block design completions and related graph decompositions.

Contribution

It provides an exact minimum number of blocks for uncompletable partial (n,k,1)-designs when n is large, generalizing Evans' conjecture.

Findings

Exact minimum number of blocks for uncompletable partial designs.

Extension of Evans' conjecture to block designs.

Results on decomposing almost complete graphs into K_k.

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 $K_k$.