  It is well known that any $N$ orthogonal pure states can always be perfectly
distinguished under local operation and classical communications (LOCC) if
$(N-1)$ copies of the state are available [Phys. Rev. Lett. 85, 4972 (2000)].
It is important to reduce the number of quantum state copies that ensures the
LOCC distinguishability in terms of resource saving and nonlocality strength
characterization. Denote $f_r(N)$ the least number of copies needed to LOCC
distinguish any $N$ orthogonal $r$-partite product states. This work will be
devoted to the estimation of the upper bound of $f_r(N)$. In fact, we first
relate this problem with Ramsey theory, a branch of combinatorics dedicated to
studying the conditions under which orders must appear. Subsequently, we prove
$f_2(N)\leq \lceil\frac{N}{6}\rceil+2$, which is better than $f_2(N)\leq
\lceil\frac{N}{4}\rceil$ obtained in [Eur. Phys. J. Plus 136, 1172 (2021)] when
$N>24$. We further exhibit that for arbitrary $\epsilon>0$,
$f_r(N)\leq\lceil\epsilon N\rceil$ always holds for sufficiently large $N$.
