A Sarvate–Beam Triple System SB(v, 3) is a set V of v elements and a collection of 3–subsets of V such that each distinct pair of elements in V occurs in i blocks, for every i in the list 1, 2, . . . , ( v 2 ) . In this paper, we completely enumerate all Sarvate-Beam Triple Systems for v = 5 and v = 6. (In the case v = 5, we extend a previous result of R. Stanton [8].) 1 Sarvate–Beam Designs Th...