Agrawal (1964) and Shah (1965) have derived bounds for the number of common treatments between the blocks of PBIB designs. Agrawal's result is more general and is applicable to any $(v, b, r, k)$ design. In this note it is established that Agrawal's bounds are superior to those given by Shah.
"Comparison of the Bounds of the Number of Common Treatments between Blocks of Certain Partially Balanced Incomplete Block Designs." Ann. Math. Statist. 37 (3) 739 - 740, June, 1966. https://doi.org/10.1214/aoms/1177699473