Lower bounds for coverings of pairs by large blocks (Q910408)

Let C(n,k,t) be the minimum cardinality of a class of k-tuples of X such that every t-tuple of X is contained in at least one k-tuple (n\(\geq k\geq t)\). Let \[ L(n,k,t)=\lceil \frac{n}{k}\lceil \frac{n-1}{k- 1}\lceil...\lceil \frac{n-t+1}{k-t+1}\rceil...\rceil \rceil \rceil \] where \(\lceil\). \(\rceil\) is the greatest integer function. Clearly C(n,k,t)\(\geq L(n,k,t)\). In this note it is shown that this lower bound can be improved in some cases.