CN-TST 2012 Problem 16

Given an integer $n\ge 4$ . $S=\{1,2,\ldots,n\}$ . $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that $$\min \{|A|,|B|\}\le\log _2n.$$