The objective of this course is to provide an in-depth and rigorous exploration of advanced combinatorial theory and techniques. Students will engage with sophisticated topics such as advanced enumeration methods, the probabilistic method, algebraic and geometric combinatorics, extremal combinatorics, combinatorial designs, and graph theory at a higher level of abstraction. The course emphasizes formal proof techniques, critical analysis of combinatorial structures, and the development of original problem-solving strategies. Upon completion, students will be able to conduct research-level investigations in combinatorics and apply combinatorial reasoning to complex problems in mathematics, theoretical computer science, and related disciplines.