Research on the existence of utility functions? I've been having difficulty finding some research on indifference curves/utility functions and methods in constructing them in an applied manner. 

Does anyone have any papers or pdf books to recommend?
The most famous one is probably Afriat's 1967 paper titled "The construction of a utility function from demand data". 

You can also find the basic idea in Hal Varian's Intermediate Microeconomics: Revealed Preferences Chapter or you can read the generalized version of the theorem in the paper.