Let $m_1\le \cdots \le m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex class sizes $m_1,\dots ,m_k$ then prove that$$R(T,G)\le (\chi (G)-1)(R(T,K_{m_1,m_2})-1)+m_1.$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let (m_1\le \cdots \le m_k) and let [ G=K_{m_1,m_2,\dots,m_k} ] be the complete $k$-partite graph with vertex-class sizes (m_1,\dots,m_k). Then (\chi(G)=k), and the smallest color class in a proper $k$-coloring of $G$ has size [ \sigma(G)=m_1. ]

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.