Combinatorics discrete solution:combinatorial object combinatorial≈ finite constraint:combinatorial structure Enumeration (counting): How many solutions satisfying the constraints? Existence: Does there exist a solution? Extrema: How large/small a solution can be to preserve/avoid certain structure? Ramsey: When a solution is sufficiently large, some structure must emerge. ●Optimization: Find the optimal solution. Construction (design): Construct a solution.Combinatorics • Enumeration (counting): • Existence: • Extremal: • Ramsey: • Optimization: • Construction (design): How many solutions satisfying the constraints? Does there exist a solution? When a solution is sufficiently large, some structure must emerge. How large/small a solution can be to preserve/avoid certain structure? Find the optimal solution. Construct a solution. solution: combinatorial object constraint: combinatorial structure combinatorial≈ discrete finite