2026/07/08 by Yongjiang Wu, Lihua Feng
#math.CO
In a 2021 survey on Katona's circle method, Frankl conjectured that every family F⊆ 2[n] in which any two members intersect and no two members cover [n] satisfies the sharp binomial norm bound ‖\mathcal F‖n :=∑F∈\mathcal F\binomn|F|-1 ≤ (n+1)/(6). This improves the earlier estimate (n)/(4) obtained by the circle method. In this paper, we prove Frankl's conjecture and determine all extremal families. Our proof develops a continuous p-biased measure approach in place of the circle method. The intersection and union conditions lead to a sharp estimate for μp(\mathcal F)+μ1-p(\mathcal F). Integrating this estimate over p converts it directly into the desired binomial norm bound and recovers the optimal coefficient (1)/(6). This continuous averaging is the key new ingredient of the proof and also yields the characterization of all extremal families.