Kahn's flow conjecture (Friedgut et al. 2018) concerns upward flows on the Boolean algebra ![2^([n])](/images/equations/KahnsFlowConjecture/Inline1.svg)
, where ![[n]={1,...,n}](/images/equations/KahnsFlowConjecture/Inline2.svg)
for a positive
integer 
.
For nonnegative functions 
and 
on ![2^([n])](/images/equations/KahnsFlowConjecture/Inline6.svg)
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 ![h:2^([n])->R](/images/equations/KahnsFlowConjecture/Inline13.svg)
be increasing under set inclusion and antipodal, so 
for the complement 
of 
in ![[n]](/images/equations/KahnsFlowConjecture/Inline17.svg)
, and write 
. Use the Boolean
Fourier expansion normalization ![h^^(T)=2^(-n)sum_(x subset= [n])h(x)(-1)^(|T intersection x|)](/images/equations/KahnsFlowConjecture/Inline19.svg)
.
Distribute each squared Fourier coefficient 
for nonempty 
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. Here a
downset 
is a family of subsets of ![[n]](/images/equations/KahnsFlowConjecture/Inline34.svg)
closed under taking subsets.
Chvátal's conjecture asserts that every intersecting
family 
satisfies ![|I|<=max_(i in [n])|D(i)|](/images/equations/KahnsFlowConjecture/Inline36.svg)
,
where 
.
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.
