Shadows of 3-uniform hypergraphs under a minimum degree condition

We prove a minimum degree version of the Kruskal–Katona theorem: given $d\ge 1/4$andatriplesystem$F$on$n$verticeswithminimumdegreeatleast$d\binom n2$, we obtain asymptotically tight lower bounds for the size of its shadow. Equivalently, for $t\ge n\mathrm{/}2-1$, we asymptotically determine the minimum size of a graph on $n$ vertices, in which every vertex is contained in at least $\left(\genfrac{}{}{0px}{}{t}{2}\right)$ triangles. This can be viewed as a variant of the Rademacher–Tur'an problem.