$f$-Diophantine sets over finite fields via quasi-random hypergraphs from multivariate polynomials

We investigate $f$-Diophantine sets over finite fields via new explicit constructions of families of quasi-random hypergraphs from multivariate polynomials. In particular, our construction not only offers a systematic method for constructing quasi-random hypergraphs but also provides a unified framework for studying various hypergraphs arising from multivariate polynomials over finite fields, including Paley sum hypergraphs, and hypergraphs derived from Diophantine tuples and their generalizations. We derive an asymptotic formula for the number of $k$-Diophantine $m$-tuples, answering a question of Hammonds et al., and study some related questions for $f$-Diophantine sets, extending and improving several recent works. We also sharpen a classical estimate of Chung and Graham on even partial octahedrons in Paley sum hypergraphs.