trisets

Trisets are sets containing exactly three elements. In standard mathematical language, a 3-element subset of a finite set S is typically called a 3-subset or a 3-element subset; the term trisets is informal and may refer either to such subsets or to the collection of all of them drawn from a larger set.

If S has size n, the number of trisets that can be formed from S is the binomial coefficient C(n,3) = n(n−1)(n−2)/6. For any fixed pair of distinct elements from S, the number of trisets containing that pair is n−2, since the third element may be chosen from the remaining n−2 elements. When n < 3, there are no trisets.

Example: for S = {a, b, c, d}, the trisets are {a, b, c}, {a, b, d}, {a, c, d}, and {b, c, d}.

In a broader combinatorial context, a collection of trisets on a ground set S is a 3-uniform hypergraph, where each edge corresponds to a 3-element subset. This viewpoint leads to problems such as 3-set packing (selecting disjoint trisets) and 3-set covering (covering S with trisets). Environments that use trisets include sampling, design of experiments, and various algorithms in combinatorial optimization.

Because trisets are not always given as a formal term in mathematics, many texts prefer the explicit phrase “3-element subsets” or “3-subsets.” See also 3-subset, binomial coefficient, and 3-uniform hypergraph.