The following two results are proved.(1) Let D be a triangle free quasi-symmetric design with k = 2y − x and x ≥ 1 then D is a trivial design with v = 5 and k = 3.(2) There do no exist triangle free quasi-symmetric designs with x ≥ 1 and λ = y or λ = y − 1.
Triangle-free quasi-symmetric 2-(v, k,k) designs with intersection numbers x, y; 01, are investigated. It is proved that k ≥ 2 y − x −3. As a consequence it is seen that for fixed k, there are finitely many triangle-free quasi-symmetric designs. It is also proved that: k ≤ y( y − x)+ x. q
A~TRACT. The following result is proved: Let D be a quasi-symmetric 3-design with intersection numbers x, y(0 ~< x < y < k). D has no three distinct blocks such that any two of them intersect in x points if and only if D is a Hadamard 3-design, or D has a parameter set (v, k, 2) where v = (2 + 2)(22 + 42 + 2) + 1, k = 22 + 32 + 2 and 2 = 1, 2 ..... or D is a complement of one of these designs.
Quasi-symmetric designs with intersection numbers x > 0 and y = x + 2 under the condition λ > 1 are investigated. If D (v, b, r, k, λ; x, y) is a quasi-symmetric design with above conditions then it is shown that either λ = x + 1 or x + 2 or D is a design with the parameters given in the Table 6 or complement of one of these designs.
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