发表机构
Department of Mathematics, University of Manchester（曼彻斯特大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明稀疏色图（边数至多 $cr^3\log^2 r$）必含 $\binom r3$ 个三角形，通过稀疏核、阶均匀稳定性和稠密公共调色板列表方法给出完整证明。
AI 中文摘要
我们证明存在绝对常数 $c>0$，使得每个色数至少为 $r$ 且边数至多为 $cr^3\log^2 r$ 的图至少包含 $\binom r3$ 个三角形。证明包含三个要素。首先，基于 Harris 的三角形敏感着色估计的稀疏核论证，从任何反例中提取一个阶为 $O(r)$ 且色数至少为 $(1-\eta)r$ 的诱导子图。其次，我们证明一个阶均匀稳定性定理：对于每个 $\beta>0$，存在 $\gamma>0$（与线性阶界中的常数无关），使得每个足够大的、阶为 $O(s)$ 且满足 $\omega(J)\le (1-\beta)s$ 的 $s$-临界图 $J$ 至少包含 $\binom s3+\gamma s^3$ 个三角形。该证明结合了过剩方法和 Fox、Tidor 与 Zhang 的修正独立集论证。第三，当图包含阶至少为 $(1-\delta)r$ 的团时，我们证明精确界。这使用了 Harris 边-三角形估计的新的稠密公共调色板列表类似物：如果每个列表占据公共调色板的固定正比例，则边-三角形着色界在常数因子内仍然成立。
英文摘要
We prove that there is an absolute constant $c>0$ such that every graph of chromatic number at least $r$ and at most $cr^3\log^2 r$ edges contains at least $\binom r3$ triangles. The proof has three ingredients. First, a sparse-core argument based on a triangle-sensitive coloring estimate of Harris extracts, from any counterexample, an induced subgraph of order $O(r)$ and chromatic number at least $(1-η)r$. Second, we prove an order-uniform stability theorem: for every $β>0$ there is $γ>0$, independent of the constant in the linear order bound, such that every sufficiently large $s$-critical graph $J$ of order $O(s)$ with $ω(J)\le (1-β)s$ has at least $\binom s3+γs^3$ triangles. The proof combines the excess method and the modified-independent-set argument of Fox, Tidor, and Zhang. Third, we prove the exact bound when the graph contains a clique of order at least $(1-δ)r$. This uses a new dense common-palette list analogue of Harris's edge--triangle estimate: if every list occupies a fixed positive proportion of a common palette, then the edge-triangle coloring bound survives up to a constant factor.
