Latin and semi-Latin factorizations of complete graphs and support sizes of quadruple systems

paper) we introduce notions of Latin and semi-Latin factorizations and their support sizes. We determine the set support of Latin and factorizations of Utilizing r!{yt"'T'l'Yl1np the set QSS(8m, A) of support sizes of quadruple systems of order 8m and index A for m 6 with at most 5 OITllSSlOIlS for each m 0 Introd uction Let be a finite set and be a We denote by the set of all k-subsets of i:)U'DP()Se that B1 and two collections of the elements of Pk(X) and m is a positive collection of the elements of B1 and B2 will be denoted by B1 + B2 and m copies of 8 1 is denoted by mBl. The set of distinct elements of B1 is called the support of B1 and is denoted. The number b'" = IBil is called the support size of B 1 • Let Xl and X 2