How do mathematicians use lower and upper bounds in combinatorics?

In combinatorics, mathematicians use lower and upper bounds primarily when exact enumeration of configurations—such as graphs, designs, or arrays—proves too difficult to calculate precisely. By establishing these bounds, researchers can approximate the number of possible configurations or narrow down the range in which a specific value must exist.1

Bounds are also critical in addressing problems of existence and optimization. For instance, when the exact value of a number is unknown, such as with certain Ramsey numbers, mathematicians provide bounds to constrain the possibilities (e.g., $35\le N_2(4,6)\le 41$).2

Furthermore, extremal properties are used to establish the existence of specific objects. By identifying an extremal figure—such as a parallelogram with the least possible area—mathematicians can prove the existence of configurations that satisfy desired properties. Conversely, researchers may use these methods in reverse, attempting to disprove the existence of an extremal figure to demonstrate that a specific condition must hold, as seen in solutions to problems in projective geometry.3