同态与VC维阈值：谱与分离
Homomorphism and VC-dimension thresholds: spectra and separations
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中文总结 AI 辅助
该研究揭示同态与VC维阈值具无限谱且非单调，证明四类全局简单性阈值两两不同，确定多部图VC维阈值及同态阈值的下界与聚点。
中文摘要 AI 辅助
最小度阈值研究的是排除固定图H时，稠密图何时能具有简单全局描述。对于每个固定色数，色阈值仅有三个可能值。我们证明这种有限谱现象是色阈值特有的：仅在3色图中，同态阈值和VC维阈值就具有无限谱，且在取诱导子图时非单调。对于含单顶点部分的完全三部图，我们证明δ_hom(K_{1,s,t}) ≥ max{1/3, s/(1+s+t)}，且在s、t的无限范围内取等；特别地，对每个s≥2，δ_hom(K_{1,s,s})=s/(2s+1)。更一般地，对每个r≥3，值(r-2)/(r-1)是r色图同态阈值的一个聚点。对于极大无H图，我们确定了每个完全三部图的VC维阈值，并证明对每个非二部图H，其VC维阈值为正，特别得到每个奇环的精确值。我们还在VC维先验约束下对色阈值进行分类。结合已知的爆破阈值结果，我们的定理表明δ_χ、δ_hom、δ_VC和δ_B两两不同：有界色性、同态可压缩性、邻域复杂度和精确爆破结构是四种本质不同的全局简单性形式。证明过程发展了有界同态像的随机与网格障碍、在极大完备化下保持高VC维的饱和装置，以及在保持无H性的同时提高最小度的核心定向方法。
英文摘要
Minimum-degree thresholds ask when excluding a fixed graph $H$ forces a dense graph to admit a simple global description. For each fixed chromatic number, the chromatic threshold has only three possible values. We show that this finite-spectrum phenomenon is special to chromatic threshold: already among $3$-chromatic graphs, both the homomorphism and VC-dimension thresholds have infinite spectra and are nonmonotone under taking induced subgraphs. For complete tripartite graphs with a singleton part, we prove $δ_{\mathrm{hom}}(K_{1,s,t}) \ge \max\left\{\frac13,\frac{s}{1+s+t}\right\}$, with equality for an infinite range of $s,t$; in particular, $δ_{\mathrm{hom}}(K_{1,s,s})=s/(2s+1)$ for every $s\ge2$. More generally, for every $r\ge3$, the value $(r-2)/(r-1)$ is an accumulation point of the homomorphism thresholds of $r$-chromatic graphs. For maximal $H$-free graphs, we determine the VC-dimension threshold of every complete tripartite graph and prove that it is positive for every nonbipartite $H$, yielding in particular the exact value for every odd cycle. We also classify the chromatic threshold under an a priori VC-dimension bound. Together with known blowup-threshold results, our theorems reveal that $δ_χ,δ_{\mathrm{hom}},δ_{\mathrm{VC}}$, and $δ_{\mathrm B}$ are \emph{pairwise distinct}: bounded colorability, homomorphic compressibility, neighborhood complexity, and exact blowup structure are genuinely different forms of global simplicity. The proofs develop random and grid-based obstructions to bounded homomorphic images, saturated gadgets that preserve high VC-dimension under maximal completion, and a core-orientation method for raising minimum degree while preserving $H$-freeness.
