AI 中文总结
该研究肯定回答了Chartrand与Zhang关于目标拉姆齐数的问题，构造了连通、平面、二分的无穷多反例，还给出不连通反例的独立构造，揭示了TR(G)/R(G)比值的无界性。
AI 中文摘要
Chartrand和Zhang提出：是否存在无孤立顶点的图G，其目标拉姆齐数满足TR(G) > R(G)？我们肯定回答了该问题，甚至在强结构限制下也成立。若Γₜ = Pₜ□Pₜ为t×t方格网，Chartrand和Zhang的初等计数界给出TR(Γₜ) ≥ 2t(t-1)+1，而Mota、Sarkozy、Schacht和Taraz的定理给出R(Γₜ) = (3/2 + o(1))t²。因此对所有足够大的t，TR(Γₜ) > R(Γₜ)，故存在无穷多个最大度为4的连通、平面、二分反例。高维方格网表明，在连通二分图上，TR(G)/R(G)的比值无界，而每个固定维数的见证族有界最大度。我们还从Burr、Erdos和Spencer关于多份拷贝拉姆齐数的定理中记录了一种不连通反例的独立构造：若固定图H满足e(H) > 2v(H) - α(H)，则对每个足够大的q，TR(qH) > R(qH)。
英文摘要
Chartrand and Zhang asked whether there exists a graph $G$ without isolated vertices whose target Ramsey number satisfies $\mathrm{TR}(G) > \mathrm{R}(G)$. We answer the question affirmatively, even under strong structural restrictions. If $Γ_t = P_t \square P_t$ is the square $t \times t$ grid, then the elementary counting bound of Chartrand and Zhang gives $\mathrm{TR}(Γ_t) \geq 2t(t-1)+1$, whereas a theorem of Mota, Sarkozy, Schacht and Taraz gives $\mathrm{R}(Γ_t) = (3/2 + o(1))t^2$. Consequently, $\mathrm{TR}(Γ_t) > \mathrm{R}(Γ_t)$ for all sufficiently large $t$, so there are infinitely many connected, planar, bipartite counterexamples of maximum degree four. Higher-dimensional grids show that the ratio $\mathrm{TR}(G)/\mathrm{R}(G)$ is unbounded on connected bipartite graphs, while each fixed-dimensional witnessing family has bounded maximum degree. We also record an independent construction of disconnected counterexamples from the theorem of Burr, Erdos and Spencer on Ramsey numbers of multiple copies: if a fixed graph $H$ satisfies $e(H) > 2v(H) - α(H)$, then $\mathrm{TR}(qH) > \mathrm{R}(qH)$ for every sufficiently large $q$.
Comments6 pages. Answers a question of Chartrand and Zhang on target Ramsey numbers
