A simple EULA does not absolve you from legal responsibility. The law that you need to be acquainted with, if you are dealing with the US (i.e. might be sued in the US), is the Digital Millennium Copyright Act, in particular Title II, the Online Copyright Infringement Liability Limitation Act which states the "safe harbor" provisions. Aspects of DMCA safe harbor are covered in many Law SE questions. In essence, you have to provide a way for rights holders to complain that someone has infringed their copyright on there site, and you have to take down allegedly infringing material: and there are a number of legal formalities to attend to in doing this. The main point is that you can't just ignore the problem and hope it goes away, and you can't just say it is not your responsibility, which is what a simple EULA does.
To be protected, you need a "designated agent" where complainers can contact you. You provide the information online (as well as stating the DMCA policy, which can be in the EULA), and also register that information with the Copyright office (online). The complaint has to be in writing, and most of the burden is on the author of the complaint, but you still have to be sure that the complaint is legally conforming. The complaint has to say what was infringed (e.g. the URL), the identity of the protected content (title of the book, for instance), and provide the complainer's signature and contact information. It also requires the complainer to say that they have a good faith belief that the material is illegally copied (no permission, and not otherwise allowed by law), and a perjury statement that the foregoing is accurate and authorized by the copyright holder.
When you have a conforming notice, you must "expeditiously" remove / disable the infringing material (there is no definition of "expeditious"), notify the user, then wait for a proper counter-claim (same general form as the take-down claim but where the user denies the posting the material was illegal. If you get a counter-claim, you notify the alleged copyright owner and wait for them to file suit in 10 days. If they don't do that, you restore the material. Here is a sample complaint, and a sample counter notice. Also, this document (look for the download tab) reorganizes the legal language so that requirements are put in logical order and not randomly scattered throughout the US Code.