An improved upper bound for the size of the multiplicative 3-Sidon sets

We say that a set is a multiplicative 3-Sidon set if the equation $s_1{s}_2{s}_3=t_1{t}_2{t}_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of $\{1,2,\dots ,n\}$ is at most $\pi(n)+\pi(n/2)+n^{2/3}(\log n )^{2^{1/3}-1/3+o(1)}$,whichimprovesthepreviouslyknownbestbound$\pi(n)+\pi(n/2)+cn^{2/3}\log n/\log\log n$.