{"paper":{"title":"Random Van der Waerden Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ohad Zohar","submitted_at":"2020-06-09T17:21:26Z","abstract_excerpt":"In this paper we prove the Random Van der Waerden Theorem: For $q_1 \\geq q_2 \\geq \\dotsb \\geq q_r \\geq 3 \\in \\mathbb{N}$ there exist $c,C >0$ such that \\[ \\lim_{n \\to \\infty} \\mathbb{P}([n]_p \\rightarrow (q_1,\\dotsc, q_r)) = \\begin{cases} 1 & \\text{if } p \\geq C \\cdot n^{-\\frac{q_2}{q_1(q_2-1)}}, 0 & \\text{if } p \\leq c \\cdot n^{-\\frac{q_2}{q_1(q_2-1)}}, \\end{cases}\\] extending the results of R\\\"odl and Ruci\\'nski for the symmetric case $q_i = q$. The proof for the 1-statement is based on the Hypergraph Container Method by Balogh, Morris and Samotij and Saxton and Thomason. The proof for the 0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.05412","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.05412/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}