Erdős–Ko–Rado for three sets

Fix integers k⩾3 and n⩾3k/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|⩽2k, we have A∩B∩C≠∅. We prove that with equality only when ⋂F∈FF≠∅. This settles a conjecture of Frankl and Füredi [2], who proved the result for n⩾k2+3k.