The crossword solutions for the term KEEP UP WITH are STAY (4 letters) and RIVAL (5 letters). To keep up with something means to maintain the same level or pace as it, and this phrase is often used in the context of competition. To STAY with something means to remain with it, while a RIVAL is someone who competes against another. Therefore, the phrase KEEP UP WITH can be interpreted as maintaining the same level or pace as a competitor.