Memoirs of the American Mathematical Society 2013; 83 pp; softcover Volume: 221 ISBN10: 0821875574 ISBN13: 9780821875575 List Price: US$62 Individual Members: US$37.20 Institutional Members: US$49.60 Order Code: MEMO/221/1038
 Let \(\bf\Gamma\) be a Borel class, or a Wadge class of Borel sets, and \(2\!\leq\! d\!\leq\!\omega\) be a cardinal. A Borel subset \(B\) of \({\mathbb R}^d\) is potentially in \(\bf\Gamma\) if there is a finer Polish topology on \(\mathbb R\) such that \(B\) is in \(\bf\Gamma\) when \({\mathbb R}^d\) is equipped with the new product topology. The author provides a way to recognize the sets potentially in \(\bf\Gamma\) and applies this to the classes of graphs (oriented or not), quasiorders and partial orders. Table of Contents  Introduction
 A condition ensuring the existence of complicated sets
 The proof of Theorem 1.10 for the Borel classes
 The proof of Theorem 1.11 for the Borel classes
 The proof of Theorem 1.10
 The proof of Theorem 1.11
 Injectivity complements
 Bibliography
