Frankston–Kahn–Narayanan regular 3-wise intersection problem

The problem asks whether every regular increasing three-wise intersecting family of subsets becomes negligible compared with the full power set as the dimension grows. Frankston, Kahn, and Narayanan answered this question in work addressing a 1989 question of Cameron, Frankl, and Kantor.

Known results

• Frankston, Kahn, and Narayanan (2017): $|\mathcal{A}|=o(2^n)$ for every regular increasing $3$-wise intersecting $\mathcal{A}\subseteq\mathcal{P}_n$, using Friedgut’s junta theorem.
• Their paper asked for substantially stronger quantitative bounds, such as $\log_2|\mathcal{A}|\leq n-cn^\delta$ for universal $c,\delta>0$.
• Frankl’s construction shows that regularity without increasingness does not force negligible size.

August 2026 quantitative strengthening

Chang and Fan claim the explicit inequality $\log(2^n/|\mathcal{A}|)\geq \frac{n}{2}\left(\frac{|\mathcal{A}|}{2^n-|\mathcal{A}|}\right)^2$, hence $|\mathcal{A}|\leq 2^n\sqrt{\mathrm{W}(n)/n}$. Their note gives a direct proof avoiding Friedgut’s theorem and says ChatGPT 5.6 assisted only with the Lambert-$W$ formulation and exposition; the mathematical argument is attributed to the authors.

Current status (as of August 2026): the qualitative $o(2^n)$ theorem is settled, while Chang and Fan’s explicit bound for regular increasing $3$-wise intersecting families remains an unverified preprint claim.