An improvement on covering triples by quadruples

✍ Scribed by L. Ji

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 140 KB
• Volume: 16
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20156

✦ Synopsis

Abstract

Let C(3,4,v) be the minimum number of four‐element subsets (called blocks or quadruples) of a v‐element set X, such that each three‐element subset of X is contained in at least one block. Let $L(3,4,v)=\lceil {v\over 4}\lceil{v-1\over 3}\lceil{v-2\over 2}\rceil\rceil\rceil$. Schönheim has obtained C(3,4,v) ≥  L(3,4,v). Further, Mills showed that C(3,4,v) = L(3,4,v) for v ≢ 7 (mod 12), and Hartman, Mills, and Mullin showed that C(3,4,v) = L(3,4,v) for v ≡ 7 (mod 12) with v ≥  52423 or v = 499. In this article, it is proved that C(3,4,v) = L(3,4,v) for all v ≡ 7 (mod 12) with an exception v = 7 and possible exceptions of v = 12__k__ + 7, k ∈ {1,2,3,4,5,7,8,9,10,11,12,16,21,23,25,29}. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 231–243, 2008