{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T08:29:56Z","timestamp":1769156996649,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Eberhard and Pohoata conjectured that every $3$-cube-free subset of $[N]$ has size less than $2N\/3+o(N)$. In this paper we show that if we replace $[N]$ with $\\mathbb{Z}_N$ the upper bound of $2N\/3$ holds, and the bound is tight when $N$ is divisible by $3$ since we have $A=\\{a\\in \\mathbb{Z}_N:a\\equiv 1,2\\pmod{3}\\}.$ Inspired by this observation we conjecture that every $d$-cube-free subset of $\\mathbb{Z}_N$ has size less than $(d-1)N\/d$ where $N$ is divisible by $d$, and we show the tightness of this bound by providing an example $B=\\{b\\in\\mathbb{Z}_N:b\\equiv 1,2,\\ldots,d-1\\pmod{d}\\}$. We prove the conjecture for several interesting cases, including when $d$ is the smallest prime factor of $N$, or when $N$ is a prime power.\r\nWe also discuss some related issues regarding $\\{x,dx\\}$-free and $\\{x,2x,\\ldots,dx\\}$-free sets. A main ingredient we apply is to arrange all the integers into some square matrix, with $m=d^s\\times l$ having the coordinate $(s+1,l-\\lfloor l\/d\\rfloor)$. Here $d$ is a given integer and $l$ is not divisible by $d$.<\/jats:p>","DOI":"10.37236\/14052","type":"journal-article","created":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T16:52:42Z","timestamp":1769100762000},"source":"Crossref","is-referenced-by-count":0,"title":["On Cube-Free Problems"],"prefix":"10.37236","volume":"33","author":[{"given":"Yuchen","family":"Meng","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,1,23]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p16\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T16:52:42Z","timestamp":1769100762000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,23]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/14052","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,23]]},"article-number":"P1.16"}}