发表机构
Zuse Institute Berlin, Technische Universität Berlin; Department of Mathematics, Duke University（柏林齐泽研究所，柏林工业大学; 杜克大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明每个足够大的素数阶 Paley 图是分形图，并构造了第二个显式 Cayley 分形图无限族，解决了极值组合学中显式构造问题。
AI 中文摘要
可诱导性问题要求在所有具有给定顶点数的图中，找出某个固定图的诱导副本的最大数量。可诱导性一直是极值组合学中一个活跃的研究领域，但对于显式定义的图模式，确定所有极值图仍然具有挑战性，特别是当描述要求在所有宿主阶数下都成立时。一个图 $H$ 被称为分形图，如果对于每个正整数 $n$，每个最大化 $H$ 的诱导副本数量的 $n$ 顶点图都是 $H$ 的平衡迭代膨胀图，该膨胀图通过递归地在大小相差至多 1 的部分中重复相同模式而获得。先前的概率结果表明，大型随机图和随机阿贝尔 Cayley 图是分形图的概率趋于 1，从而确立了它们的丰富性，但没有直接提供显式族。我们证明每个足够大的素数阶 Paley 图都是分形图。因此，这些经典算术模式决定了每个极值宿主的精确递归结构，在所有宿主阶数下且不要求宿主具有任何代数假设。我们还构造了第二个显式的非平凡 Cayley 分形图无限族，对于该族，分形图性质比 Paley 情形有更简单的证明。这些结果共同解决了 2025 年美国数学研究所“旗代数与极值组合学”研讨会上讨论的显式构造问题。
英文摘要
The inducibility problem asks for the maximum number of induced copies of a fixed graph among all graphs with a prescribed number of vertices. Inducibility has been an active area of research in extremal combinatorics, but determining all extremal graphs for explicitly defined patterns remains challenging, particularly when the description is required to hold at every host order. A graph $H$ is called a fractalizer if, for every positive integer $n$, every $n$-vertex graph maximizing the number of induced copies of $H$ is a balanced iterated blow-up of $H$, obtained by recursively repeating the same pattern in parts whose sizes differ by at most one. Previous probabilistic results show that large random graphs and random abelian Cayley graphs are fractalizers with probability tending to one, establishing their abundance without directly providing explicit families. We prove that every sufficiently large prime-order Paley graph is a fractalizer. Thus these classical arithmetic patterns determine the exact recursive structure of every extremal host, at every host order and without any algebraic assumptions on the host. We also construct a second explicit infinite family of nontrivial Cayley fractalizers, for which the fractalizer property admits a simpler proof than in the Paley case. Together, these results resolve the explicit-construction question discussed at the 2025 American Institute of Mathematics workshop "Flag Algebras and Extremal Combinatorics."
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