Poonen's conjecture and Ramsey numbers

For cQ, let c:QQ denote the quadratic map c(X)=X2+c. How large can the period of a rational periodic point of c be? Poonen conjectured that it cannot exceed 3. Here, we tackle this conjecture by graph-theoretical means with the Ramsey numbers Rk(3). We show that, for any cQ whose denominator admits at most k distinct prime factors, the map c admits at most 2Rk(3)2 periodic points. As an application, we prove that Poonens conjecture holds for all cQ whose denominator is a power of 2.