On the Existence of S(3,{4,5,7},v)

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• 作者：Lijun Ji and Yun Li
• 刊名：Journal of Combinatorial Designs
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：25
• 期：3
• 页码：122-143
• 全文大小：235K
• ISSN：1520-6610

文摘

Let B3(K)={v: there is an S(3,K,v)}. For K={4} or {4,6}, B3(K) has been determined by Hanani, and for K={4,5} or {4,5,6}, B3(K) has been determined by the first author. In this paper, we investigate the case K={4,5,7}. A necessary condition for v∈B3({4,5,7}) is v≡1,2(mod3). It is known that B3({4,5})={v≥4:v≡1,2,4,5,8,10(mod12),v≠13}⊂B3({4,5,7}), and that there is an S(3,{4,5,7},v) for all v≡7(mod12) with a possible exception v=19. We need to consider the case v≡11(mod12). It is proved that there is an S(3,{4,5,7},v) for all v≡11(mod12) with an exception v=11 and a possible exception v=23, thereby, B3({4,5,7})∪{19,23}={v≥4:v≡1,2(mod3),v≠11,13}.