Shadows of 3-Uniform Hypergraphs under a Minimum Degree Condition

Abstract

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