{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:42:49Z","timestamp":1753893769278,"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>A line in $({\\Bbb Z}\/n{\\Bbb Z})^2$ is any translate of a cyclic subgroup of order $n$.  A subset $X\\subset ({\\Bbb Z}\/n{\\Bbb Z})^2$ is a cap if no three of its points are collinear, and $X$ is complete if it is not properly contained in another cap. We determine bounds on $\\Phi(n)$, the minimum size of a complete cap in $({\\Bbb Z}\/n{\\Bbb Z})^2$. The other natural extremal question of determining the maximum size of a cap in $({\\Bbb Z}\/n{\\Bbb Z})^2$ is considered in a separate preprint by the present author.  These questions are closely related to well-studied questions in finite affine and projective geometry.  If $p$ is the smallest prime divisor of $n$, we prove that $$\\max\\{4,\\sqrt{2p}+{1\\over2}\\}\\leq \\Phi(n)\\leq \\max\\{4,p+1\\}.$$ We conclude the paper with a large number of open problems in this area.<\/jats:p>","DOI":"10.37236\/1084","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:59:34Z","timestamp":1578718774000},"source":"Crossref","is-referenced-by-count":1,"title":["The Minimum Size of Complete Caps in $({\\Bbb Z}\/n{\\Bbb Z})^2$"],"prefix":"10.37236","volume":"13","author":[{"given":"Jack","family":"Huizenga","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2006,7,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v13i1r58\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v13i1r58\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:10:25Z","timestamp":1579320625000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v13i1r58"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,7,28]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2006,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1084","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2006,7,28]]},"article-number":"R58"}}