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In this chapter, the author describes several problems he have worked on over the years which are still mostly unresolved. This paper is based on a talk on this subject which he presented at the 50th Southeastern Conference on Combinatorics, Graph Theory and Computing held in Boca Raton on March 4 - 8, 2019. The paper was on what we called universal cycles for combinatorial structures. These are ways of efficiently representing classes of combinatorial objects in the form of a cycle, with the various combinatorial objects appearing uniquely as a “window” of fixed width moves around the cycle. Binomial coefficients have been the source of innumerable number-theoretic problems since they were first identified, which according to some accounts dates back to the second century B.C. In general, the middle binomial coefficients tend to be highly composite. The Minimum Spanning Tree problem is a classic topic …