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arXiv · 2608.08679

Forcing Quasirandomness via Rooted F-Densities

Heng Li·Xizhi Liu
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Abstract

Let $F$ be a finite graph with at least one edge, and let $W$ be a graphon. We show that if the density of $F$ rooted at each edge is almost everywhere constant, then either $t(F,W)=0$ or $W$ is constant. For edge-transitive $F$, one rooted equation suffices. This recovers the edge-rooted triangle theorem of Reiher and Schacht. In their terminology, our result also shows that every clique is $2$-forcing, answering a question they posed. We give an explicit stability estimate when $W$ is bounded away from zero. Our proof has two steps: an entropy argument turns constant rooted densities into an additive identity for $\log W$, and a Hoeffding decomposition determines all solutions of that identity. The same method gives exact classifications and quantitative stability estimates for symmetric uniform hyperkernels, dissociated Aldous--Hoover hypergraphons, directed kernels, and tournamentons.

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Heng Li, Xizhi Liu. 2026-08-09. Forcing Quasirandomness via Rooted F-Densities. https://arxiv.org/abs/2608.08679

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