Shadows of 3-Uniform Hypergraphs under a Minimum Degree Condition

We prove a minimum degree version of the Kruskal--Katona theorem for triple systems: given $d\ge 1/4$ and system $\mathcal{F}$ on $n$ vertices with $\delta(\mathcal{F})\ge d\binom n2$, we obtain asymptotically tight lower bounds size its shadow. Equivalently, $t\ge n/2-1$, determine graph vertices, in which every vertex is contained at least $\binom t2$ triangles. This can be viewed as variant Rademacher--Turán problem.