**Update: This is a third of three posts (part I, part II) proposing some extensions of the cap set problem and some connections with the Frankl Rodl theorem. Here is a post presenting the problem on Terry Tao’s blog (March 2007). Here is an open discussion post around Roth theorem (March 2009). Here are two “open discussion” posts on the cap set problem (first, and second) from Gowers’s blog (January 2011). These posts include several interesting Fourier theoretic approaches towards improvement of the Roth-Meshulam bound. The second post describes a startling example by Seva Lev.**

Suppose that the maximum size of a cap set in is . Here is an obvious fact:

The maximum size of a set in with the property that every () satisfy is at most the maximum size of a cap set in .

**Proof:** Indeed the condition for is stronger than being a cap set. We require for every not only that but even .

## Part C. A more direct relation between Frankl-Rodl’s theorem and the cap set problem and all sorts of fine gradings on subsets of {1,2,…,n}.

In part A we moved from sets avoiding affine lines (cap sets) to sets avoiding “modular lines” and used the Frankl-Rodl theorem to deal with the latter. This may look artificial. Here is a simpler connection between the cap set problem and the Frankl-Rodl theorem.

### 17. Why weaker forms of the Frankl-Rodl Theorem follow from upper bounds on cap sets.

Consider Continue reading