If a family \(\mathcal {F}\) of k-element subsets of an n-element set is intersecting, then the sum of the sizes of the pairwise intersections of any \(\ell \) members of the family is at least \({\ell \atopwithdelims ()2}\) . The classic result of Erdős, Ko and Rado says that under the condition \(2k\le n\) an intersecting family of k-element subsets of an n-element set cannot have more than \({n-1\atopwithdelims ()k-1}\) members. Is this weaker condition for the sum of the sizes of the pairwise intersections sufficient to have the conclusion of the Erdős–Ko–Rado theorem? We will see that much more is true, the bound \({\ell \atopwithdelims ()2}\) can be replaced by \({\ell -1\atopwithdelims ()2}+1\) .
