JOURNAL OF COMBINATORIAL THEORY (A) 12, 304-308 (1972) Note A Characterization of Symmetric Block Designs* JACK E. GRAVER Syracuse University, Syracuse, N. Y. 13210 Communicated by H. J. Ryser Received September 2, 1970 By a system of blocks we shall mean a finite indexed collection S, ,..., S, of subsets of a finite set S = {a1 ,..., a, }. ki will denote ISi I, the cardinality of the i-th block; ri = 1 {i 1 S, 3 uj} 1 is called the j-th replication number. If i # j, htf = / SC n Sj 1 is called the i, j intersection parameter. The system is said to be a symmetric block design if the following conditions are satisfied: (0) 12 = m; (1) ki = k, for 1 < i < n; (1’) rj = r, for 1 ,C j < m; (2) Aij = X,for 1 <icj<n. (This definition includes the degenerate systems: (a) Si = S for all i, (b) & = q5 for all i, (c) Si = {a,(s} for some permutation Z- of (I,..., n>.) There are standard theorems to the effect that: any system satisfying (0), (l), (2) with h -=c k is a symmetric block design and that any system satisfying (0), (l’), (2) with h > 0 is a symmetric block design. In [2] Ryser shows that there are systems other than symmetric block designs satisfying (0) and (2) but neither (1) nor (1’). He calls these systems &designs. For i # j we may define Q = I Si + Sj I (where + is Boolean sum: S, + Sj = {a ) a E Si u Sj , a 4 & n S,}) which we will call the i, j sum parameter. A symmetric block design obviously also satisfies: (3) oij=a,forl<iij<n. * This research was supported in part by NSF grant GP-19404. 304 0 1972 by Academic Press, Inc. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector