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 λ ( υ,k ) for infinitely many values of υ, k and λ , including infinitely many values of υ and k for λ =1. Here, C λ ( υ,k ) denotes the minimum number of k -subsets of a υ -set of points such that each pair of points occurs in at least λ 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.