AI 中文总结
研究有界 VC 维数图的齐次集，改进 Nguyen 等人方法，给出更精确量化界\(\eta_d\geq (Cd)^{-d}\)，通过直接应用 VC 维数假设优化归纳，还推导多种相关图类的定量结果。
AI 中文摘要
图中的齐次集是团或稳定集。Erdős--Hajnal 猜想指出，对于每个图\(H\)，存在\(c>0\)，使得每个\(n\)个顶点的不含\(H\)的图都有一个大小至少为\(n^c\)的齐次集。Nguyen、Scott 和 Seymour 证明，对于每个\(d>0\)，VC 维数至多为\(d\)的图具有 Erdős--Hajnal 性质，证实了 Fox、Pach 和 Suk 的一个猜想。本文给出了 VC 维数至多为\(d\)的图中齐次集的更精确量化界，表明可以取\(\eta_d\geq (Cd)^{-d}\)，其中\(C\)是一个绝对常数。我们的证明改进了 Nguyen、Scott 和 Seymour 的迭代稀疏化方法，主要增强是直接应用 VC 维数假设，从而得到更有效的归纳并改善了对\(d\)的依赖。我们还推导了多项式 Rödl 子图、有界 VC 维数下的超图 Ramsey 界、诱导自由和病毒公式、竞赛图、NIP 和半代数图、有界 VC 维数关系的布尔组合、邻接矩阵秩有界的图、有界符号秩的图以及由点积阈值表示定义的图的定量结果。
英文摘要
A homogeneous set in a graph is a clique or a stable set. The Erdős--Hajnal conjecture states that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a homogeneous set of size at least $n^c$. Nguyen, Scott and Seymour proved that for every $d>0$, graphs of VC-dimension at most $d$ have the Erdős--Hajnal property, confirming a conjecture of Fox, Pach and Suk. In particular, they showed that every such $n$-vertex graph contains a homogeneous set of size at least $n^{η_d}$ for some $η_d\ge 2^{-2^{O(d)}}$. In this paper, we give a sharper quantitative bound on the homogeneous sets in graphs of VC-dimension at most $d$, showing that one may take $ η_d\ge (Cd)^{-d}, $ where $C$ is an absolute constant. Equivalently, every graph $G$ of VC-dimension at most $d$ satisfies \[ \max\{ω(G),α(G)\}\ge |G|^{(Cd)^{-d}}. \] Our proof refines the iterative sparsification method of Nguyen, Scott and Seymour. The main enhancement is to apply the VC-dimension assumption directly, which gives a more efficient induction and thus improves the dependence on $d$. We also derive quantitative consequences for polynomial Rödl subgraphs, hypergraph Ramsey bounds under bounded VC-dimension, induced-free and viral formulations, tournaments, NIP and semi-algebraic graphs, Boolean combinations of relations of bounded VC-dimension, graphs whose adjacency matrices have bounded rank, graphs of bounded sign-rank, and graphs defined by dot-product threshold representations.
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