Minimum degrees and codegrees of minimal Ramsey 3-uniform hypergraphs

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• 作者：Dennis Clemens ^{dennis.clemens@tuhh.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：minimal Ramsey hypergraph ; minimum degree and codegree
• 刊名：Electronic Notes in Discrete Mathematics
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：49
• 期：Complete
• 页码：23-30
• 全文大小：211 K

文摘

A uniform hypergraph H is called k-Ramsey for a hypergraph F, if no matter how one colors the edges of H with k colors, there is always a monochromatic copy of F. We say that H is minimal k-Ramsey for F, if H is k-Ramsey for F but every proper subhypergraph of H is not. Burr, Erdős and Lovasz [S. A. Burr, P. Erdős, and L. Lovász, On graphs of Ramsey type, Ars Combinatoria 1 (1976), no. 1, 167–190] studied various parameters of minimal Ramsey graphs. In this paper we initiate the study of minimum degrees and codegrees of minimal Ramsey 3-uniform hypergraphs. We show that the smallest minimum vertex degree over all minimal k  -Ramsey 3-uniform hypergraphs for $K_t^{(3)}$ is exponential in some polynomial in k and t. We also study the smallest possible minimum codegrees over minimal 2-Ramsey 3-uniform hypergraphs.