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.