Determines the maximum cardinality of a family of k-element subsets of an n-element set such that every pair of subsets in the family has nonempty intersection; in the classical form, when n ≥ 2k the maximum size is the number of k-subsets containing a fixed element.

Principle

An extremal-combinatorics principle: among uniform k-subset families with a pairwise-intersection constraint, extremal families are highly structured (typically stars), and counting/closure operations (shifting, compression) yield optimal bounds.

Demonstration

Example: for n = 8 and k = 3 with n ≥ 2k, the maximum size of an intersecting family is the number of 3-subsets that contain a fixed element, namely C(7,2) = 21; one constructs any maximal intersecting family by fixing an element and taking all k-sets that include it.

Misapplication

Applying the classical bound when n < 2k or to nonuniform families without modification; assuming the same extremal structure for t-intersecting families (where intersections must have size ≥ t) without using the appropriate generalization.

Consequence

When applicable, the theorem gives both a sharp numeric upper bound and a structural characterization of extremal families (stars), which guides proofs and constructions in intersecting-family problems.

Reversal

The inverse perspective considers families required to be pairwise disjoint: maximizing size under pairwise-disjointness leads to different combinatorial maxima and constructions (matchings), illustrating how flipping the intersection condition changes optimal families.

Boundary

Applies to uniform families of k-element subsets of an n-set; the classical bound requires n ≥ 2k for the star to be extremal; variants and extensions are needed for n < 2k, t-intersecting conditions, or nonuniform collections.

Semantic Tension

Competes with nearby concepts such as theorems about t-intersecting families or the Ahlswede–Khachatrian complete intersection theorem; the tension is between 'intersecting' as pairwise nonempty intersection and stronger intersection size constraints that change extremal structure.

Synthesis

Erdős–Ko–Rado quantifies and characterizes the largest possible uniform family where every pair meets: under the classical n ≥ 2k condition the largest families are those fixing a common element (stars), and combinatorial compression methods certify optimality.