Kahn’s flow conjecture on the Boolean cube
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The question is whether for every increasing antipodal function h on the Boolean cube and every nonnegative allocation of each nonempty Fourier coefficient’s squared mass among its coordinates, the induced linear function flows upward to h₊². This is Kahn’s flow conjecture.
Why it matters and what counts as progress
Why it matters: It asks for a structural flow principle strengthening the known intersecting-family inequalities for every permitted Fourier allocation.
What counts as progress: Prove upward flow for every permitted allocation of Fourier mass, beyond a single preferred allocation, or construct a violating allocation.
How the problem developed
- 2018 · Conjecture [1]
Ehud Friedgut, Jeff Kahn, Gil Kalai and Nathan Keller Formulated the flow conjecture as a strengthening related to Chvátal’s intersecting-family conjecture and Boolean-cube correlation inequalities.
- 16 Sept 2026 · Partial result [2]
Fan Chang, Hong Liu and Miao Liu Posted a sharp correlation inequality proving Chvátal’s conjecture; this result became an input to the later flow argument.
- 20 Sept 2026 · Proof claim [3]
Peter Keevash Posted a proof of Kahn’s flow conjecture, using the Chang–Liu–Liu inequality and crediting GPT-6 Astra with finding the proof within a human-suggested approach.
AI contribution
On Kahn's flow conjecture
For every increasing antipodal function h on the Boolean cube and every nonnegative allocation of each nonempty Fourier coefficient’s squared mass among its coordinates, the induced linear function flows upward to h₊². This is Kahn’s flow conjecture.
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