AI 中文总结
该研究针对无C₄图，推导了三角形密度的上下界，在团复形为2-Leray时得到更优的团数密度界，还提出了针对特定2-Leray图的推测性尖锐界及相关问题。
AI 中文摘要
对于顶点数为n的无C₄图G（即不存在由4个顶点构成的诱导环的图），我们研究三角形密度τ的双边极值问题：在给定边密度ε和团数密度κ=ω(G)/n的情况下，τ的最大和最小可能值是多少？我们给出了用κ和ε表示的τ的下界与上界，这两个界将τ夹在中间，且二者的相容性要求κ需满足用ε表示的下界。当G的团复形在域k上是2-Leray时，得到的团数密度界介于此前最优的无C₄界与尖锐弦图界之间，且对所有ε∈(0,1)均优于前者。下界是初等的，上界是同调的，通过过渡到团复形的Stanley–Reisner环得到。当该复形是2-Leray时，其Betti表至多有两条线性 strand，第一条 strand 的前两个条目编码边密度与三角形密度，而Boij–Söderberg分解的强结构形式会约束这些条目的可能取值，从而得到上界。对于在[4,g]范围内无洞的2-Leray图，我们给出了一个推测为尖锐的界，还进一步针对任意无C₄图的三角形密度提出了相关问题。
英文摘要
For a $C_4$-free graph $G$ on $n$ vertices --- one with no induced cycle on four vertices --- we study the two-sided extremal problem for the triangle density $τ$: How large and how small can $τ$ be for given edge density $\varepsilon$ and clique-number density $κ= ω(G)/n$? We give lower and upper bounds for $τ$ in terms of $κ$ and $\varepsilon$. The two bounds sandwich $τ$, and their compatibility forces a lower bound for $κ$ in terms of $\varepsilon$. When the clique complex of $G$ is $2$-Leray over a field $\Bbbk$, the resulting bound on the clique-number density lies between the previous best $C_4$-free bound and the sharp chordal bound. It improves on the former {\it for every} $\varepsilon \in (0,1)$. The lower bound is elementary. The upper bound is homological, obtained by passing to the Stanley--Reisner ring of the clique complex. When the complex is $2$-Leray, its Betti table has at most two linear strands. The two first entries in the first strand encode edge and triangle densities, and the strong structural form of a Boij--Söderberg decomposition constrains what these entries can be, yielding the upper bound. For $2$-Leray graphs with no holes in the range $[4,g]$ we give a conjecturally sharp bound. We further ask questions concerning the triangle bound for any $C_4$-free graph.
Comments25 pages, 3 figures
