subsetten

Subsetten is a term used in combinatorics to denote a family of ten-element subsets of a finite set X that satisfies a prescribed intersection pattern. The basic idea is to study how large a collection of 10-element subsets can be such that any two members meet in a fixed number t of elements.

Formal definition: Let X be a finite set with n elements. For a fixed integer t with 0 ≤ t < 10, a subsetten family F is a collection of subsets A ⊆ X with |A| = 10 for all A ∈ F, such that for any distinct A, B ∈ F, |A ∩ B| = t. When t is chosen as 0, the family is pairwise disjoint; when t > 0, the intersections are constrained to exactly t elements.

Existence and size: The parameter triple (n, 10, t) determines feasibility. If n is too small, no such nontrivial family exists beyond trivial cases. For larger n, one seeks the maximum possible |F| and explicit constructions. Techniques come from extremal set theory and design-theoretic methods, with many results depending on the exact values of n and t.

Relation to other concepts: Subsetten relates to k-set families, intersecting families, and block designs. It is distinct from classical t-designs but shares common tools, such as counting arguments and combinatorial bounds.

Examples: With n = 11 and t = 9, a nontrivial subsetten is not possible; a trivial example is restricting to a single ten-element block, giving |F| = 1. For larger n and smaller t, more complex constructions may exist.

History and usage: Subsetten is a term used primarily in theoretical discussions and educational contexts to illustrate fixed-size subset arrangements and intersection constraints. See also: k-sets, intersecting families, block designs.