Kahn's Flow Conjecture

Kahn's flow conjecture (Friedgut et al. 2018) concerns upward flows on the Boolean algebra , where . For nonnegative functions and on having the same sum, flows upward to if there are nonnegative numbers supported on pairs whose row sums are and whose column sums are .

Let be increasing and antipodal, so for the complement of in , and write $h_+(x)=max{h(x),0}$ . Distribute each squared nonempty Fourier coefficient among the coordinates using nonnegative weights satisfying , and put

			(1)
			(2)

The conjecture asserts that flows upward to . Keevash (2026) proved the conjecture, from which the Chvátal conjecture on intersecting subfamilies of downsets follows.

Keevash (2026) reports that GPT-6 Astra found the proof by following his proposed approach, after which he simplified and rewrote the argument. As of Sep. 22, 2026, independent specialist review had not been reported.

References

Chvátal, V. "Intersecting Families of Edges in Hypergraphs Having the Hereditary Property." In Hypergraph Seminar (Ed. C. Berge and D. Ray-Chaudhuri). Lecture Notes in Mathematics, Vol. 411. Berlin, Germany: Springer-Verlag, pp. 61-66, 1974.

Friedgut, E.; Kahn, J.; Kalai, G.; and Keller, N. "Chvátal's Conjecture and Correlation Inequalities." J. Combin. Theory Ser. A 156, 22-43, 2018. https://doi.org/10.1016/j.jcta.2018.01.001.

Keevash, P. "On Kahn's Flow Conjecture." 20 Sep 2026. https://arxiv.org/abs/2609.23595.

Kahn's Flow Conjecture

See also

Explore with Wolfram|Alpha

References

Cite this as:

Subject classifications