{"paper":{"title":"Capturing Forms in Dense Subsets of Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Brandon Hanson","submitted_at":"2012-11-25T16:09:32Z","abstract_excerpt":"An open problem of arithmetic Ramsey theory asks if given a finite $r$-colouring $c:\\mathbb{N}\\to\\{1,...,r\\}$ of the natural numbers, there exist $x,y\\in \\mathbb{N}$ 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 $\\mathbb{F}_q$. Specifically, given a linear form $L$ and a quadratic from $Q$ in two variables, we provide estimates on the necessary size of $A\\subset \\mathbb{F}_q$ to guarantee that $L(x,y)$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5771","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}