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Consider the following problem. In a school with three classes containing n students each, given that their genders are unknown, find the minimum possible number of triples of same-gender students—not all of which are from the same class. Muaengwaeng asked this question and conjectured that the minimum scenario occurs when the classes are all boys, all girls and half-and-half. In this paper, we solve many generalizations of the problem including when the school has more than three classes, when triples are replaced by groups of larger sizes, when the classes are of different sizes, and when gender is replaced by other non-binary attributes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C (r , k , v) of order v is a collection of r-uniform k-cycles on a v-element vertex set, such that each r-element subset is an edge in precisely one of those k-cycles. We present cyclic hypercycle systems C (3 , 5 , v) of orders v = 25 , 26 , 31 , 35 , 37 , 41 , 46 , 47 , 55 , 56 , a highly symmetric construction for v = 40 , and cyclic 2-split constructions of orders 32 , 40 , 50 , 52 . As a consequence, all orders v ≤ 60 permitted by the divisibility conditions admit a C (3 , 5 , v) system. New recursive constructions are also introduced. [ABSTRACT FROM AUTHOR]