Poonen’s conjecture and Ramsey numbers

For c∈Q∗c∈Q∗, let φc:Q→Qφc:Q→Q denote the quadratic map φc(X)=X2+cφc(X)=X2+c. How large can the period of a rational periodic point of φcφc be? Poonen conjectured that it cannot exceed 3. Here, we tackle this conjecture by graph-theoretical means with the Ramsey numbers Rk(3)Rk(3). We show that, for any c∈Q∗c∈Q∗ whose denominator admits at most kk distinct prime factors, the map φcφc admits at most 2Rk(3)−22Rk(3)−2 periodic points. As an application, we prove that Poonen’s conjecture holds for all c∈Q∗c∈Q∗ whose denominator is a power of 2.