kuniforms

Kuniforms, in mathematics, refer to k-uniform hypergraphs. A hypergraph H = (V, E) is k-uniform if every edge e ∈ E contains exactly k vertices, i.e., |e| = k for all edges. The vertex set V has size n, and the edge set E is a collection of k-element subsets of V. The complete k-uniform hypergraph on n vertices, denoted K_n^{(k)}, contains all possible k-subsets as edges, so |E| = C(n, k).

When k = 2, kuniforms reduce to ordinary graphs, with the degree of a vertex defined as the number of incident edges. In general, properties such as degrees, codegrees (numbers of edges containing a given pair of vertices), and incidence structures are central to the study of kuniforms. Counting, extremal, and probabilistic questions extend naturally from graphs to k-uniform hypergraphs.

Variations and related concepts include linear k-uniform hypergraphs, where any two edges intersect in at most one vertex, and balanced incomplete block designs, which are k-uniform hypergraphs with each pair of vertices appearing in exactly λ edges. These structures connect to combinatorial design theory and have applications in experimental design and information theory.

Theoretical topics commonly studied for kuniforms include random models, such as the G^{(k)}(n, p) model where each k-subset of V becomes an edge with probability p, and extremal problems, including Turán-type questions and the Erdős–Ko–Rado theorem on intersecting families. Counting problems are also prominent, with the number of labeled kuniform hypergraphs on n vertices equal to 2^{C(n, k)}.

Applications of kuniforms span modeling multi-way relationships in data, network science, coding theory, and design of experiments, where higher-order interactions exceed pairwise connections.