An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field . Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size of to guarantee that L(x,y) and Q(x,y) are elements of A for some .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-3-4,
author = {Brandon Hanson},
title = {Capturing forms in dense subsets of finite fields},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {277-284},
zbl = {1316.11011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-3-4}
}
Brandon Hanson. Capturing forms in dense subsets of finite fields. Acta Arithmetica, Tome 161 (2013) pp. 277-284. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-3-4/