AI 中文总结
本文针对有限图F与图on W，证明F的边有根密度几乎处处为常数时W的性质，恢复了边根三角形定理，给出团为2-强制的结论及稳定性估计，方法可推广至多类核结构。
AI 中文摘要
设F为至少含一条边的有限图，W为图on。我们证明：若F在每条边处的有根密度几乎处处为常数，则要么t(F,W)=0，要么W为常数。对边传递的F，仅需一个有根方程即可。这可恢复Reiher与Schacht的边根三角形定理。按其术语，我们的结果还表明每个团都是2-强制的，回答了他们提出的一个问题。当W有界且远离零时，我们给出显式稳定性估计。我们的证明分两步：熵论证将常数有根密度转化为log W的加性恒等式，Hoeffding分解确定该恒等式的所有解。同一方法可对对称均匀超核、解离的Aldous--Hoover超图on、定向核及锦标赛on给出精确分类与定量稳定性估计。
英文摘要
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.
