{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T19:14:13Z","timestamp":1760382853790,"version":"3.41.2"},"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>Let $m$ be a positive integer whose smallest prime divisor is denoted by $p$, and let ${\\Bbb Z}_m$ denote the cyclic group of residues modulo $m$. For a set $B=\\{x_1,x_2,\\ldots,x_m\\}$ of $m$ integers satisfying $x_1 &lt; x_2 &lt; \\cdots &lt; x_m$, and an integer $j$ satisfying $2\\leq j \\leq m$, define $g_j(B)=x_j-x_1$. Furthermore, define $f_j(m,2)$ (define $f_j(m,{\\Bbb Z}_m)$) to be the least integer $N$ such that for every coloring $\\Delta : \\{1,\\ldots,N\\}\\rightarrow \\{0,1\\}$ (every coloring $\\Delta : \\{1,\\ldots,N\\}\\rightarrow {\\Bbb Z}_m$), there exist two $m$-sets $B_1,B_2\\subset \\{1,\\ldots,N\\}$ satisfying: (i) $\\max(B_1) &lt;  \\min(B_2)$, (ii) $g_j(B_1)\\leq g_j(B_2)$, and (iii) $|\\Delta (B_i)|=1$ for $i=1,2$ (and (iii) $\\sum_{x\\in B_i}\\Delta (x)=0$ for $i=1,2$). We prove that $f_j(m,2)\\leq 5m-3$ for all $j$, with equality holding for $j=m$, and that $f_j(m,{\\Bbb Z}_m)\\leq 8m+{m\\over p}-6$. Moreover, we show that $f_j(m,2)\\ge 4m-2+(j-1)k$, where $k=\\left\\lfloor\\left(-1+\\sqrt{{8m-9+j\\over j-1}}\\right){\/2}\\right\\rfloor$, and, if $m$ is prime or $j\\geq{m\\over p}+p-1$, that $f_j(m,{\\Bbb Z}_m)\\leq 6m-4$.  We conclude by showing $f_{m-1}(m,2)=f_{m-1}(m,{\\Bbb Z}_m)$ for $m\\geq 9$.<\/jats:p>","DOI":"10.37236\/1054","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:03:03Z","timestamp":1578700983000},"source":"Crossref","is-referenced-by-count":6,"title":["Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter"],"prefix":"10.37236","volume":"13","author":[{"given":"David","family":"Grynkiewicz","sequence":"first","affiliation":[]},{"given":"Rasheed","family":"Sabar","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2006,3,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v13i1r28\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v13i1r28\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:17:49Z","timestamp":1579303069000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v13i1r28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,3,30]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2006,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1054","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2006,3,30]]},"article-number":"R28"}}