Introduction to linear-algebraic method
The theme of this section is as follows: given a family with certain restrictions on pairwise intersections, we associate polynomials with each set and show that they are linearly independent. We then bound the size of the family by the dimension of the space of the polynomials.
Summary: the oddtown theorem, the Frankl-Wilson theorem.
For bigger picture see Extremal Set Theory Course
Take a family such that the sizes of all sets in the family are odd/even and their intersections are odd/even. How big can it be? The answer is dramatically different depending on the choice of the conditions we make. Here is the video by Tim Gowers:
Take a prime p and a family of 2p-element sets in [4p] with no two sets intersecting in exactly p elements. How big can this family be? The upper bound is given by the Frankl-Wilson theorem, which is very useful in several problems in combinatorial geometry. Here is the video by Tim Gowers: