Vector space ramsey numbers and weakly Sidorenko affine configurations.

This article focuses on affine extremal and Ramsey-type problems in vector spaces over finite fields, particularly improving bounds on Ramsey numbers related to affine subspaces. It introduces the concept of affine extremal numbers, which measure the largest subsets of |$\mathbb{F}_q^n$| avoiding affine copies of a given configuration, and establishes new supersaturation results by counting affine homomorphisms. The authors improve upper bounds on two-color Ramsey numbers |$R_q(s,t)$| for |$q \in \{2,3\}$|, including exponential improvements for off-diagonal cases |$R_2(2,t)$| and |$R_3(2,t)$|, by connecting these problems to affine extremal numbers and leveraging the Sidorenko property of certain affine configurations. The work also reformulates off-diagonal Ramsey problems as affine extremal problems involving direction sets and provides unified proofs of key extremal bounds over |$\mathbb{F}_2$| and |$\mathbb{F}_3$|, while posing open questions about the general behavior of affine configurations over arbitrary finite fields.