Ramsey Number of Disjoint Union of Good Hypergraphs

Abstract

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.