Erdős Problem #537

Let \(\epsilon>0\) and \(N\) be sufficiently large. If \(A\subseteq \{1,\ldots,N\}\) has \(\lvert A\rvert \geq \epsilon N\) then must there exist \(a_1,a_2,a_3\in A\) and distinct primes \(p_1,p_2,p_3\) such that \[a_1p_1=a_2p_2=a_3p_3?\] A positive answer would imply [536]. Erdős describes a construction of Ruzsa which disproves this: consider the set of all squarefree numbers of the shape \(p_1\cdots p_r\) where \(p_{i+1}>2p_i\) for \(1\leq i<r\). This set has positive density, and hence if \(A\) is its intersection with \((N/2,N)\) then \(\lvert A\rvert \gg N\) for all large \(N\). Suppose now that \(p_1a_1=p_2a_2=p_3a_3\) where \(a_i\in A\) and \(p_1,p_2,p_3\) are distinct primes. Without loss of generality we may assume that \(a_2>a_3\) and hence \(p_2<p_3\), and so since \(p_2p_3\mid a_1\in A\) we must have \(2<p_3/p_2\). On the other hand \(p_3/p_2=a_2/a_3\in (1,2)\), a contradiction.