By J. A. Bondy (auth.), D. A. Holton, Jennifer Seberry (eds.)

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Additional info for Combinatorial Mathematics: Proceedings of the International Conference on Combinatorial Theory Canberra, August 16–27, 1977

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As H e K(l,l,m)/t, such that H'~l H for each i. l of the points u I and u 2 respectively. Since u I and u 2 each have degree m+l in K(A,B,C), the sum over the t graphs of these degrees is t (1) there is some t [ dil = Z di2 =m+l. i=l i=l 51 Let r be the m a x i m u m degree among the points of H. i, so by (i) we have tr a m+l. Then dil ~ r for each As t > 1 and m e t(t+l), m/t > 2 and hence r > 2. Let v be a point in H of degree r. Then for each i, the point of H. associated with l 2 in K(A,B,C).

T-designs Bt[k, l; v] exist Similarly the set of integers v for which balanced will be denoted by Bt(k, ~). The following lemmas are evident. 1• K c Bt(K , i). 2• If K' c K, then Bt(K' , X) c Bt(K , A). 3. If I' divides I, then Bt(K , I') c Bt(K , l). 4. Bt(K, I) n Bt(K , I') c Bt(K , n~+n'X'), where ~ and I' are any positive integers, and n and n' any non-negative integers. 5. If v ~ Bt(K' , I') and K' c Bt(K , I), then v ~ Bt(K, II') holds. 5 will be most useful. 6. If v £ Bt(K , I) and K c Bt(k , I), then v ~ Bt(k, I) holds.

TRANSVERSAL then S + i = {s+l : s c S}. t-DESIGNS Let s ~ t, r and k be positive A transversal t-design Tt[s , k; r] is points, g - a family of subsets of X called groups, which form a partition of X, and P a family of subsets of X called blocks, a triple (X, G, P), where satisfying Ioi[ (i) IGI (ii) (iii) conditions: = r for every G i E = @; s; IGi 0 Bjl = I for every G i E @ and every Bj c P; (iv) group, the following integers. X is a set of every t-subset is contained It follows [Bjl = s of X, such that each of its points in exactly immediately in a distinct of P.

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