{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:09Z","timestamp":1753893849605,"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>\u00a0Let $d\\in\\mathbb{Z}^+t$, $\\mathbb{K}$ be a field of characteristic zero and $A$ be a nonempty finite subset of $\\mathbb{K}^2$. Denote by $\\mathcal{C}_{d,\\mathbb{K}}$ the family of algebraic curves of degree $d$ in $\\mathbb{K}^2$ and $\\mathcal{C}_{\\leq d,\\mathbb{K}}:=\\bigcup_{e=1}^d\\mathcal{C}_{e,\\mathbb{K}}$. For any $C_1\\in \\mathcal{C}_{d,\\mathbb{K}}$, we say that $C_1$ is determined by $A$ if for any $C_2\\in\\mathcal{C}{d,\\mathbb{K}}$ such that $C_2\\cap A\\supseteq C_1\\cap A$, we have that $C_1=C_2$; we denote by $\\mathcal{D}_{d,\\mathbb{K}}(A)$ the family of elements of $\\mathcal{C}_{d,\\mathbb{K}}$ determined by $A$. Beck's theorem establishes that if $\\mathbb{K}=\\mathbb{R}$ and $A$ is not collinear, then $$|\\mathcal{D}_{1,\\mathbb{R}}(A)|=\\Theta\\left(|A|\\min_{C\\in \\mathcal{C}_{1,\\mathbb{R}}}|A\\setminus C|\\right).$$ In this paper we generalize Beck's theorem showing that for all $d\\in\\mathbb{Z}^+$, there exists a constant $c=c(d)&gt;0$ such that if\u00a0 $\\min_{C\\in\\mathcal{C}_{\\leq d,\\mathbb{K}}}|A\\setminus C|&gt;c,$ then\u00a0\u00a0$$|\\mathcal{D}_{d,\\mathbb{K}}(A)|=\\Theta_d\\left(|A|^d\\prod_{e=1}^d\\left(\\min_{C\\in \\mathcal{C}_{\\leq e,\\mathbb{K}}}|A\\setminus C|\\right)^{d-e+1}\\right).$$<\/jats:p>","DOI":"10.37236\/9658","type":"journal-article","created":{"date-parts":[[2020,12,24]],"date-time":"2020-12-24T01:44:25Z","timestamp":1608774265000},"source":"Crossref","is-referenced-by-count":0,"title":["Beck's Theorem for Plane Curves"],"prefix":"10.37236","volume":"27","author":[{"given":"Mario","family":"Huicochea","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2020,12,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i4p54\/8235","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i4p54\/8235","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,24]],"date-time":"2020-12-24T01:44:25Z","timestamp":1608774265000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i4p54"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,24]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,10,2]]}},"URL":"https:\/\/doi.org\/10.37236\/9658","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,12,24]]},"article-number":"P4.54"}}