{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:33Z","timestamp":1753893813255,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In this paper, we are concerned with calculating $r(k, n)$, the length of the longest $k$-AP free subsequences in $1, 2, \\ldots , n$. We prove the basic inequality $r(k, n) \\le n \u2212 \\lfloor m\/2\\rfloor$, where $n = m(k \u2212 1) + r$ and $r &lt; k \u2212 1$. We also discuss a generalization of a famous conjecture of Szekeres (as appears in\u00a0Erd\u0151s and Tur\u00e1n) and describe a simple greedy algorithm that appears to give an optimal $k$-AP free sequence infinitely often. We provide many exact values of $r(k, n)$ in the Appendix.<\/jats:p>","DOI":"10.37236\/3007","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:12:17Z","timestamp":1578705137000},"source":"Crossref","is-referenced-by-count":0,"title":["Unique Sequences Containing No $k$-Term Arithmetic Progressions"],"prefix":"10.37236","volume":"20","author":[{"given":"Tanbir","family":"Ahmed","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Janusz","family":"Dybizb\u00e1nski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hunter","family":"Snevily","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2013,12,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p29\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:10:37Z","timestamp":1579259437000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i4p29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,12,17]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2013,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/3007","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,12,17]]},"article-number":"P29"}}