An improvement on covering triples by quadruples

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