On the non-existence of pair covering designs with at least as many points as blocks

We establish new lower bounds on the pair covering number C (lambda) (upsilon,k) for infinitely many values of upsilon, k and lambda, including infinitely many values of upsilon and k for lambda=1. Here, C (lambda) (upsilon,k) denotes the minimum number of k-subsets of a upsilon-set of points such that each pair of points occurs in at least lambda of the k-subsets. We use these results to prove simple numerical conditions which are both necessary and sufficient for the existence of (K (k) - e)-designs with more points than blocks

More nonexistence results for symmetric pair coverings

A $(v,k,\lambda)$-covering is a pair $(V, \mathcal{B})$, where $V$ is a $v$-set of points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ (called blocks), such that every unordered pair of points in $V$ is contained in at least $\lambda$ blocks in $\mathcal{B}$. The excess of such a covering is the multigraph on vertex set $V$ in which the edge between vertices $x$ and $y$ has multiplicity $r_{xy}-\lambda$, where $r_{xy}$ is the number of blocks which contain the pair $\{x,y\}$. A covering is symmetric if it has the same number of blocks as points. Bryant et al.(2011) adapted the determinant related arguments used in the proof of the Bruck-Ryser-Chowla theorem to establish the nonexistence of certain symmetric coverings with $2$-regular excesses. Here, we adapt the arguments related to rational congruence of matrices and show that they imply the nonexistence of some cyclic symmetric coverings and of various symmetric coverings with specified excesses.Comment: Submitted on May 22, 2015 to the Journal of Linear Algebra and its Application