{"paper":{"title":"A Constructive Lower Bound on Szemer\\'edi's Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Vladislav Taranchuk","submitted_at":"2017-11-11T18:54:52Z","abstract_excerpt":"Let $r_k(n)$ denote the maximum cardinality of a set $A \\subset \\{1,2, \\dots, n \\}$ such that $A$ does not contain a $k$-term arithmetic progression. In this paper, we give a method of constructing such a set and prove the lower bound $n^{1-\\frac{c_k}{k \\ln k}} < r_k(n)$ where $k$ is prime, and $c_k \\rightarrow 1$ as $k \\rightarrow \\infty$. This bound is the best known for an increasingly large interval of $n$ as we choose larger and larger $k$. We also demonstrate that one can prove or disprove a conjecture of Erd\\H{o}s on arithmetic progressions in large sets once tight enough bounds on $r_k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.04183","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"}