Poonen's conjecture and Ramsey numbers

9th International Colloquium on Graph Theory and Combinatorics (ICGT), Grenoble, FRANCE, JUN 30-JUL 04, 2014International audienceFor c is an element of Q*, let phi(c) : Q -> Q denote the quadratic map phi(c)(X) = X-2 + c. How large can the period of a rational periodic point of phi(c) be? Poonen conjectured that it cannot exceed 3. Here, we tackle this conjecture by graph-theoretical means with the Ramsey numbers R-k(3). We show that, for any c is an element of Q* whose denominator admits at most k distinct prime factors, the map phi(c) admits at most 2R(k)(3) - 2 periodic points. As an application, we prove that Poonen's conjecture holds for all c is an element of Q* whose denominator is a power of 2. (C) 2015 Elsevier B.V. All rights reserved