MathDBKahn's equal-size upward-closed systems conjecture
Kahn's equal-size upward-closed systems conjecture
Let be the power set of ordered by inclusion, and let be upward closed set systems of equal size. Define the set of points belonging to exactly one of the three systems by
Kahn's conjecture. If , then
The conjecture is motivated by the product construction for three independent upward-closed systems. The source gives an upper bound of for equal-sized systems and notes that the conjectured bound is false in the preceding stronger formulation for ; it does not state whether this equal-size formulation is resolved.
Progress summary
A 2025 preprint claims a counterexample, so the conjecture appears false, but the claim has not been independently verified.
Kahn's conjecture asserts that three upward-closed systems of equal size have density at most of points belonging to exactly one system.
February 2025 counterexample claim
A preprint dated February 20, 2025, claims a counterexample in dimension , with exactly-one density exceeding , hence exceeding . Its construction starts from a weighted-hypercube example in dimension and lifts it while preserving equal densities. The result is presented as a disproof, but the source is an unverified arXiv preprint rather than an independently corroborated proof. The stronger bound was already known to fail for ; that is a different formulation.
Current status (as of August 2026): A preprint claims to disprove Kahn's equal-size conjecture in dimension , but absent independent verification the conjecture is not settled; related componentwise questions remain open.
Sources
Sources & referencesView supporting material
Primary source
Kada Williams, “A Correlation Inequality on Three Functions”, arXiv:2502.14857 (2025).
Solutions 1
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Disproven by Kada Williams in 2025, see "A Correlation Inequality on Three Functions" (https://doi.org/10.48550/arXiv.2502.14857).