TITLE: Spencer's "six standard deviations" theorem - better constants?
QUESTION [24 upvotes]: This question is about Joel Spencer's famous "six standard deviations" theorem. The theorem says that when 
$$
L_i(x_1,\dots,x_n) = a_{i1} x_1 + \dots + a_{in} x_n, \quad 1 \leq i \leq n,
$$
are $n$ linear forms in $n$ variables with all $|a_{ij}| \leq 1$, then there exist numbers $\varepsilon_1,\dots,\varepsilon_n \in \{-1,+1\}$ such that
$$
|L_i(\varepsilon_1,\dots,\varepsilon_n)|  \leq K \sqrt{n}
$$
for all $i$.
It is stated as Theorem 1 in: 
Spencer, Joel. Six standard deviations suffice. Trans. Amer. Math. Soc. 289 (1985), no. 2, 679–706. Full text PDF (open access)
As noted at the end of the paper, the constant $K$ which Spencer obtained is actually $K=5.32$.
Question: does anybody know of a proof of the Theorem which gives a smaller value for the constant?

REPLY [18 votes]: Having done the original result it is exciting to see work on the constant.  My 5.32 was certainly not meant to be best possible.  For the lower bounds, using
Hadamard Matrices (with coefficients +1,-1 as opposed to the set system with 
coefficients 0,1) gives a lower bound of K=1.  -- js