Sharpening of the Erdős–Ko–Rado theorem

Abstract If a family $$\mathcal {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> of k -element subsets of an n -element set is intersecting, then the sum of the sizes of the pairwise intersections of any $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> members of the family is at least $${\ell \atopwithdelims ()2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mfrac> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mfenced> </mml:math> . The classic result of Erdős, Ko and Rado says that under the condition $$2k\le n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> an intersecting family of k -element subsets of an n -element set cannot have more than $${n-1\atopwithdelims ()k-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mfrac> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> </mml:mfenced> </mml:math> 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 $${\ell \atopwithdelims ()2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mfrac> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mfenced> </mml:math> can be replaced by $${\ell -1\atopwithdelims ()2}+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mfrac> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mfenced> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> .