发表机构
School of Mathematical Sciences, Huaqiao University; Academy of Plateau Science and Sustainability and School of Mathematics and Statistics, Qinghai Normal University; Department of Mathematics and Statistics, Wright State University（华侨大学数学科学学院; 青海师范大学高原科学与可持续发展研究院及数学与统计学院; 赖特州立大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过统一直接证明确认了Alon等人关于星形图list Ramsey数的猜想，并确定了双星和细分星在显式参数下的Ramsey数，填补了已知界的空白。
AI 中文摘要
对于图\\(H\\)，\\(k\\)-色Ramsey数\\(r(H;k)\\)是使得\\(K_N\\)的每个\\(k\\)-边着色都包含一个单色的\\(H\\)副本的最小整数\\(N\\)。一个\\(k\\)-list分配满足每条边\\(|L(e)|=k\\)。list Ramsey数\\(r_\ell(H;k)\\)是使得在\\(E(K_N)\\)上存在一个\\(k\\)-list分配，且从这些list中的每个着色都包含一个单色的\\(H\\)副本的最小整数\\(N\\)。令\\(K_{1,n}\\)为星，\\(S(n,m)\\)为通过连接\\(K_{1,n}\\)和\\(K_{1,m}\\)的中心得到的双星，\\(S_n^m\\)为从\\(K_{1,n}\\)通过将\\(m\\)条边细分一次得到的图。Alon等人猜想对所有\\(k,n\ge1\\)，\\(r_\ell(K_{1,n};k)=r(K_{1,n};k)\\)。在本文中，我们通过统一的直接证明，对所有\\(k\ge1\\)和\\(n\ge3\\)确认了他们的猜想。对于偶数\\(k\ge4\\)且在显式参数条件下，我们证明了当\\(n\\)为偶数时\\(r(S(n,m);k)=kn+m+2\\)，当\\(n\\)为奇数时\\(r(S_n^m;k)=k(n-1)+m+2\\)。对于两种颜色，我们建立了一个一般的list Ramsey下界，并在显式参数范围内确定了双星和细分星的Ramsey数与list Ramsey数的共同值。这些结果填补了已知界中的几个空白。
英文摘要
For a graph \(H\), the \(k\)-color Ramsey number \(r(H;k)\) is the least integer \(N\) such that every \(k\)-edge-coloring of \(K_N\) contains a monochromatic copy of \(H\). A \(k\)-list assignment has \(|L(e)|=k\) for every edge. The list Ramsey number \(r_\ell(H;k)\) is the least integer \(N\) for which there exists a \(k\)-list assignment on \(E(K_N)\) such that every coloring from the lists contains a monochromatic copy of \(H\). Let \(K_{1,n}\) be a star, \(S(n,m)\) the double star obtained by joining the centers of \(K_{1,n}\) and \(K_{1,m}\), and \(S_n^m\) the graph obtained from \(K_{1,n}\) by subdividing \(m\) edges once. Alon et al.\ conjectured that \(r_\ell(K_{1,n};k)=r(K_{1,n};k)\) for all \(k,n\ge1\). In this paper, we confirm their conjecture for all \(k\ge1\) and \(n\ge3\) by a unified direct proof. For even \(k\ge4\) and under explicit parameter conditions, we prove that \(r(S(n,m);k)=kn+m+2\) for even \(n\) and \(r(S_n^m;k)=k(n-1)+m+2\) for odd \(n\). For two colors, we establish a general list Ramsey lower bound and determine the common values of Ramsey and list Ramsey numbers for double and subdivided stars in explicit parameter ranges. These results close several gaps in the known bounds.
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