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Ramsey Number of Disjoint Union of Good Hypergraphs
Iranian Journal of Science and Technology, Transactions A: Science ( IF 1.4 ) Pub Date : 2020-05-11 , DOI: 10.1007/s40995-020-00872-w
Azam Kamranian , Ghaffar Raeisi

For given r-uniform hypergraphs \({\mathcal {G}}\) and \({\mathcal {H}}\), the Ramsey number \(R({\mathcal {G}},{\mathcal {H}};r)\) is defined as the smallest positive integer p such that any red/blue coloring of the edges of the complete r-uniform hypergraph on p vertices contains either a red copy of \({\mathcal {G}}\) or a blue copy of \({\mathcal {H}}\). The hypergraph \({\mathcal {G}}\) is called \({\mathcal {H}}\)-good, if \(R({\mathcal {G}},{\mathcal {H}};r)=(|V({\mathcal {G}})|-1) (\chi ({\mathcal {H}})-1)+s({\mathcal {H}}),\) where \(s({\mathcal {H}})\) is the chromatic surplus number of \({\mathcal {H}}\), i.e., the minimum cardinality of color classes over all colorings of \(V({\mathcal {H}})\) with \(\chi ({\mathcal {H}})\) colors. In this note, for a given r-uniform hypergraph \({\mathcal {H}}\), the exact value of the Ramsey number of the disjoint union of \({\mathcal {H}}\)-good hypergraphs versus \({\mathcal {H}}\) will be determined exactly.



中文翻译:

良好超图的不相交联盟的Ramsey数

对于给定的r一致超图\({\ mathcal {G}} \)\ {{\ mathcal {H}} \},拉姆西数\(R({\ mathcal {G}},{\ mathcal {H }}; r)\)定义为最小的正整数p,以使p个顶点上完整r-一致超图的边缘的任何红色/蓝色着色都包含\({\ mathcal {G}}的红色副本\)\({\ mathcal {H}} \)的蓝色副本。超图\({\ mathcal {G}} \)称为\({\ mathcal {H}} \}-好,如果\(R({\ mathcal {G}},{\ mathcal {H}}; r)=(| V({\ mathcal {G}})| -1)(\ chi({\ mathcal {H}})-1)+ s({\ mathcal {H}}),\)其中\(s({{mathcal {H}})\)\({\ mathcal {H}} \)的色余数,即\(V({ \ mathcal {H}})\)具有\(\ chi({\ mathcal {H}})\)颜色。在本说明,对于给定的[R -uniform超图\({\ mathcal {H}} \) ,的不相交并的Ramsey数的精确值\({\ mathcal {H}} \) -良好对比超图\({\ mathcal {H}} \)将被完全确定。

更新日期:2020-05-11
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