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Motivated by parallel developments in ordinary and $\ell$-degree Tur\'an problems, we study the set of $(r-2)$-uniform Tur\'an densities of possibly infinite families of $r$-uniform hypergraphs. We prove that, for all sufficiently large $r$, these densities exhibit a phase transition at $4r^{-r}$, from a countable set...
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...
For an $n$-dimensional convex body $K$, let $\theta_L(K)$ denote its lattice covering density, and let $\Theta_L^{\mathrm{conv}}(n)$ and $\Theta_L^{\mathrm{sym}}(n)$ be the corresponding worst-case quantities over all convex bodies and over origin-symmetric convex bodies, respectively. Before this work, these quantitie...
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