Shadow of hypergraphs under a minimum degree condition

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

Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c(1)(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree delta(1)(G) > d then every vertex of G is contained in a copy of F in G? We asymptotically determine c(1)(n, F) when F is the generalized triangle K-4((3)), and we give close to optimal bounds in the case where F is the tetrahedron K-4((3)) (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an nvertex graph G with m> n(2)/4 edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case