某些小图的半诱导性
Semi-Inducibility of some small graphs
AI总结：
本文研究了某些小图的半诱导性，通过分析红蓝边染色完全图中H的副本数量，得出了四个和五个顶点图的精确结果，并为树和星图提供了通用结论。
摘要：
设H是一个边被染成红色和蓝色的固定图，令β∈[0,1]。令I(H, β)为在具有红色边密度为β的大型红/蓝边染色完全图G中，H的副本数量的最大值（渐近归一化）。这细化了确定H的半诱导性的问题，而后者本身是经典问题确定H的诱导性的泛化。对于β∈[0,1]的I(H, β)函数，除了当H是单色完全图（Kruskal-Katona）或单色星图（Reiher-Wagner）时外，此前没有任何图H有超过三个顶点的情况是已知的。我们为某些四个和五个顶点的图获得了精确结果，回答了多位作者提出的一些近期问题。我们还为树和星图获得了一些一般结果。仍有许多开放问题有待解决。
英文摘要：
Let $H$ be a fixed graph whose edges are colored red and blue and let $β\in [0,1]$. Let $I(H, β)$ be the (asymptotically normalized) maximum number of copies of $H$ in a large red/blue edge-colored complete graph $G$, where the density of red edges in $G$ is $β$. This refines the problem of determining the semi-inducibility of $H$, which is itself a generalization of the classical question of determining the inducibility of $H$. The function $I(H, β)$ for $β\in [0,1]$ was not known for any graph $H$ on more than three vertices, except when $H$ is a monochromatic clique (Kruskal-Katona) or a monochromatic star (Reiher-Wagner). We obtain sharp results for some four and five vertex graphs, addressing several recent questions posed by various authors. We also obtain some general results for trees and stars. Many open problems remain.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
