The Beck-Fiala conjecture asserts that there is an absolute constant 
such that every finite family 
of subsets of a finite
set 
, with each element in at
most 
members, satisfies
 |
for every positive integer 
. Here 
is the set system
discrepancy. The parameter 
bounds the number of sets containing
an element, rather than the size of each set
(Beck and Fiala 1981).
The Beck-Fiala theorem gives the classical bound 
for 
(Beck and Fiala 1981). Guo et al. (2026) report
a bound with square-root dependence
 |
It follows from their Komlós conjecture result by dividing each column of the incidence matrix,
with rows indexed by sets and columns by elements,
by 
. Every resulting column has Euclidean
norm at most 1. The constant is not asserted to be optimal, and the argument
does not establish an algorithm running in polynomial
time. When 
, every member of 
is empty and the discrepancy is zero.
