Shadows of 3-uniform hypergraphs under a minimum degree condition

TL;DR

This paper establishes a minimum degree condition for 3-uniform hypergraphs that guarantees asymptotically tight bounds on the size of their shadows, extending classical combinatorial theorems to a new setting.

Contribution

It provides a minimum degree version of the Kruskal--Katona theorem for 3-uniform hypergraphs, with asymptotically optimal bounds on shadow sizes.

Findings

Derived asymptotically tight lower bounds for shadows of hypergraphs.

Connected minimum degree conditions to triangle containment in graphs.

Extended classical combinatorial theorems to hypergraph settings.

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 $d\binom n2$, 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--Tur\'an problem.