{"publication":"Combinatorica","publisher":"Springer Nature","language":[{"iso":"eng"}],"date_updated":"2024-04-09T10:40:08Z","abstract":[{"text":"In 1916, Schur introduced the Ramsey number r(3; m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph Kn, there is a monochromatic copy of K3. He showed that r(3; m) ≤ O(m!), and a simple construction demonstrates that r(3; m) ≥ 2Ω(m). An old conjecture of Erdős states that r(3; m) = 2Θ(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.","lang":"eng"}],"publication_status":"published","quality_controlled":"1","arxiv":1,"_id":"15275","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1912.02342","open_access":"1"}],"keyword":["Computational Mathematics","Discrete Mathematics and Combinatorics"],"article_type":"original","year":"2021","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"date_created":"2024-04-03T07:59:57Z","doi":"10.1007/s00493-021-4530-9","volume":41,"author":[{"last_name":"Fox","first_name":"Jacob","full_name":"Fox, Jacob"},{"last_name":"Pach","first_name":"János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","full_name":"Pach, János"},{"last_name":"Suk","first_name":"Andrew","full_name":"Suk, Andrew"}],"type":"journal_article","department":[{"_id":"HeEd"}],"citation":{"chicago":"Fox, Jacob, János Pach, and Andrew Suk. “Bounded VC-Dimension Implies the Schur-Erdős Conjecture.” <i>Combinatorica</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00493-021-4530-9\">https://doi.org/10.1007/s00493-021-4530-9</a>.","short":"J. Fox, J. Pach, A. Suk, Combinatorica 41 (2021) 803–813.","apa":"Fox, J., Pach, J., &#38; Suk, A. (2021). Bounded VC-dimension implies the Schur-Erdős conjecture. <i>Combinatorica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00493-021-4530-9\">https://doi.org/10.1007/s00493-021-4530-9</a>","ista":"Fox J, Pach J, Suk A. 2021. Bounded VC-dimension implies the Schur-Erdős conjecture. Combinatorica. 41(6), 803–813.","ama":"Fox J, Pach J, Suk A. Bounded VC-dimension implies the Schur-Erdős conjecture. <i>Combinatorica</i>. 2021;41(6):803-813. doi:<a href=\"https://doi.org/10.1007/s00493-021-4530-9\">10.1007/s00493-021-4530-9</a>","ieee":"J. Fox, J. Pach, and A. Suk, “Bounded VC-dimension implies the Schur-Erdős conjecture,” <i>Combinatorica</i>, vol. 41, no. 6. Springer Nature, pp. 803–813, 2021.","mla":"Fox, Jacob, et al. “Bounded VC-Dimension Implies the Schur-Erdős Conjecture.” <i>Combinatorica</i>, vol. 41, no. 6, Springer Nature, 2021, pp. 803–13, doi:<a href=\"https://doi.org/10.1007/s00493-021-4530-9\">10.1007/s00493-021-4530-9</a>."},"month":"11","day":"20","article_processing_charge":"No","date_published":"2021-11-20T00:00:00Z","status":"public","oa_version":"Preprint","issue":"6","external_id":{"arxiv":["1912.02342"]},"page":"803-813","intvolume":"        41","title":"Bounded VC-dimension implies the Schur-Erdős conjecture","publication_identifier":{"eissn":["1439-6912"],"issn":["0209-9683"]}}