Erdős–Ko–Rado for three sets

Fix integers k ⩾ 3 and n ⩾ 3 k / 2 . Let F be a family of k-sets of an n-element set so that whenever A , B , C ∈ F satisfy | A ∪ B ∪ C | ⩽ 2 k , we have A ∩ B ∩ C ≠ ∅ . We prove that | F | ⩽ n - 1 k - 1 with equality only when ⋂ F ∈ F F ≠ ∅ . This settles a conjecture of Frankl and Füredi [2], who proved the result for n ⩾ k 2 + 3 k .