Shadows of 3-uniform hypergraphs under a minimum degree condition

Abstract

We prove a minimum degree version of the Kruskal--Katona theorem: given $d\ge 1/4$ and a triple system $F$ on $n$ vertices with minimum degree at least $d\left(\genfrac{}{}{0ex}{}{n}{2}\right)$, 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 $\left(\genfrac{}{}{0ex}{}{t}{2}\right)$ triangles. This can be viewed as a variant of the Rademacher--Turán problem.