I recently co-wrote a long survey (with Goaoc, Meunier, and Mustafa) about these 5 fundamental Theorems in geometric topological combinatorics. My key point will be to outline some connections between these theorems; some known, some new, some are easy some are not. Please see survey at http://front.math.ucdavis.edu/1706.05975  People who send us comments, corrections and suggestions, will be rewarded !!