发表机构
University of Illinois Urbana–Champaign; University of Mississippi（伊利诺伊大学厄巴纳-香槟分校; 密西西比大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究无拥挤超图的分数色数，证明了其渐近上界，并推广至线性超图，改进了现有结果，同时简化了证明。
AI 中文摘要
给定任意固定整数 $k \ge 2$ 和充分大的 $d$，我们证明了 $k$-均匀 $d$-退化无拥挤超图 $H$（即围长至少为 $5$）的最大可能分数色数满足 \\[ \chi_f(H) = (1 + o_d(1)) \left((k-1)\\,\frac{d}{\log d}\right)^{\frac{1}{k-1}}。\\] 事实上，我们证明这对于围长至少为 $g$ 的 $k$-均匀 $d$-退化超图也成立，其中 $g \ge 5$ 为任意给定值。作为推论，我们获得了 $d$-退化线性超图的分数色数的改进界。这项工作建立在 Allen、Dhawan 和 Noel 最近的结果之上，将其从图推广到超图。除了克服超图环境中出现的新困难外，我们的方法即使在原始图的情况下也提供了更简单的证明。我们的上界证明使用了一个更简单的迭代过程来采样独立集。我们还建立了具有局部需求的分数着色界，这是 Kelly 和 Postle 引入的一个框架，验证了 Yu 和 Zhang 最近的一个猜想。因此，我们获得了无拥挤超图独立数的度序列界，其领先常数与破碎阈值相匹配。对于匹配的下界，我们使用均匀附着模型的超图变体和调和顶点权重，通过线性规划对偶性来界定分数色数，然后通过删除总权重可忽略的顶点来移除所有短环。我们还用矩阵范数估计取代了图情况中使用的加泰罗尼亚数论证，简化了分析。
英文摘要
Given any fixed integer $k \ge 2$ and sufficiently large $d$, we show that the largest possible fractional chromatic number of a $k$-uniform $d$-degenerate uncrowded hypergraph $H$ (i.e., with girth at least $5$) satisfies \[ χ_f(H) = (1 + o_d(1)) \left((k-1)\,\frac{d}{\log d}\right)^{\frac{1}{k-1}}. \] In fact, we prove that this holds for $k$-uniform $d$-degenerate hypergraphs of girth at least $g$, for any given $g \ge 5$. As a corollary, we obtain improved bounds on the fractional chromatic number of $d$-degenerate linear hypergraphs. This work builds upon a recent result by Allen, Dhawan, and Noel, extending it from graphs to hypergraphs. In addition to overcoming the new difficulties that arise in the hypergraph setting, our approach yields a simpler proof even in the original graph case. Our proof of the upper bound uses a simpler iterative procedure for sampling independent sets. We also establish bounds for fractional colorings with local demands, a framework introduced by Kelly and Postle, verifying a recent conjecture of Yu and Zhang. As a consequence, we obtain a degree-sequence bound on the independence number of uncrowded hypergraphs with a leading constant matching the shattering threshold. For the matching lower bound, we use a hypergraph variant of the uniform attachment model and harmonic vertex weights to bound the fractional chromatic number via linear programming duality, then remove all short cycles by deleting vertices of negligible total weight. We also replace the Catalan-number argument used in the graph case with a matrix-norm estimate, simplifying the analysis.
Comments29 pages
