The six standard deviations suffice theorem, also called Spencer's theorem, states that every family 
of 
subsets of an 
-element set
has set system discrepancy satisfying
 |
Spencer (1985) proved the result. The same bound holds for an 
matrix whose entries are
real numbers with absolute
value at most 1, with discrepancy defined as the minimum of 
over sign vectors 
. Here 
is the infinity
norm. The constant 6 is not optimal.
Reis and Song (2026) report that, for every positive integer 
and every ![A in [-1,1]^(n×n)](/images/equations/SixStandardDeviationsSufficeTheorem/Inline9.svg)
,
some sign vector 
satisfies
 |
where 
denotes the inverse
hyperbolic sine. Its leading coefficient is less than 2.9992. The additive 4
is part of the bound, so this does not assert 
for every 
.
