基于类型的随机图的正则诱导子图的Sharp渐近行为
The lower-bound problem for regular induced subgraphs of type-based random graphs
AI总结：
该研究确定了基于类型的随机图模型族中正则诱导子图最大阶数相关函数f(n)的最优渐近常数√(2e)，指出相关猜想的反例需来自非基于类型的构造。
AI中文摘要：
对于图G，令F(G)表示G的正则诱导子图的最大阶数，f(n)=min{F(G):|V(G)|=n}。Erdos、Fajtlowicz和Staton提出了一个问题：f(n)/log n是否趋向于无穷大。对f(n)上界的每一次改进都来自基于类型（graphon）的随机图模型：Bollobás、Alon-Krivelevich-Sudakov、Dyson-McKay先后得到f(n)≤√(163n/9)=4.2557√n。我们从两侧确定了该整个模型族的最优常数，证明f(n)≤(√(2e)+o(1))√n=2.3316…√n，且所有基于类型的模型几乎必然满足F(G)≥(√(2e)-o(1))√n。因此，常数√(2e)是该模型族内的最优值，对f(n)上界的任何进一步改进——尤其是对Erdos-Fajtlowicz-Staton猜想的任何反例——都必须来自非基于类型的构造。
英文摘要：
For a graph G let F(G) denote the largest order of a regular induced subgraph of G, and let f(n) = min{F(G) : |V(G)| = n}. A problem of Erdos, Fajtlowicz and Staton asks whether f(n)/log n -> infinity. Dyson and McKay have recently proved f(n) <= (sqrt(2e)+o(1)) sqrt(n) via a type-based random model (arXiv:2604.08215). This paper concerns the opposite direction within the type-based family. We conjecture that every model of the family satisfies F(G) >= (sqrt(2e)-o(1)) sqrt(n) asymptotically almost surely, so that sqrt(2e) is the optimal constant obtainable from the family, and we prove the corresponding statement at exponent level -- F(G) >= n^{1/2-eps} -- conditionally on two explicitly stated hypotheses: a local limit lower bound for inhomogeneous degree sequences, and a correlation estimate at sublinear overlaps. The complementary overlap range, including full overlap, requires no correlation hypothesis. We further record the exact-curvature first-moment computation that independently identifies sqrt(2e), including a uniform trace bound on its determinant correction, and certified exact computations at orders up to 48 consistent with the predicted crossover F ~ min(n^{2/3}, sqrt(n/L)). This version substantially revises v1; see the note in Section 1.
