{"paper":{"title":"Maximum $k$-sum $\\mathbf{n}$-free sets of the 2-dimensional integer lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Boram Park, Ilkyoo Choi, Ringi Kim","submitted_at":"2019-03-11T05:40:13Z","abstract_excerpt":"For a positive integer $n$, let $[n]$ denote $\\{1, \\ldots, n\\}$. For a 2-dimensional integer lattice point $\\mathbf{b}$ and positive integers $k\\geq 2$ and $n$, a \\textit{$k$-sum $\\mathbf{b}$-free set} of $[n]\\times [n]$ is a subset $S$ of $[n]\\times [n]$ such that there are no elements ${\\mathbf{a}}_1, \\ldots, {\\mathbf{a}}_k$ in $S$ satisfying ${\\mathbf{a}}_1+\\cdots+{\\mathbf{a}}_k =\\mathbf{b}$. For a 2-dimensional integer lattice point $\\mathbf{b}$ and positive integers $k\\geq 2$ and $n$, we determine the maximum density of a {$k$-sum $\\mathbf{b}$-free set} of $[n]\\times [n]$. This is the fir"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.04132","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"}